Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.8%

Time: 24.7s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. sin-asin 99.8%

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

   3. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

   1. fma-udef 99.8%

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

9. Applied egg-rr 99.8%

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

    1. pow1 99.8%

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

11. Applied egg-rr 99.8%

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

    1. unpow1 99.8%

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

    2. associate-*r* 99.8%

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

    3. *-commutative 99.8%

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

    4. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

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(delta) * sin(phi1)) + Float64(sin(delta) * Float64(cos(phi1) * 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) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta)))))));
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[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $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* 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. remove-double-neg 99.8%

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

   5. fma-def 99.7%

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

   6. associate-*l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in delta around inf 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 94.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. sin-asin 99.8%

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

   3. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in theta around 0 94.5%

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

9. Final simplification 94.5%

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

Alternative 4: 94.8% 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* 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. remove-double-neg 99.8%

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

   5. fma-def 99.7%

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

   6. associate-*l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in theta around 0 94.5%

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

5. Final simplification 94.5%

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

Alternative 5: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * cos(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(delta) * (sin(theta) * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * cos(phi1))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in theta around 0 94.5%

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

5. Final simplification 94.5%

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

Alternative 6: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;theta \leq -8000000 \lor \neg \left(theta \leq 0.41\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (if (or (<= theta -8000000.0) (not (<= theta 0.41)))
     (+
      lambda1
      (atan2 (* (sin theta) t_1) (- (cos delta) (pow (sin phi1) 2.0))))
     (+
      lambda1
      (atan2
       (* (sin delta) (* theta (cos phi1)))
       (- (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);
	double tmp;
	if ((theta <= -8000000.0) || !(theta <= 0.41)) {
		tmp = lambda1 + atan2((sin(theta) * t_1), (cos(delta) - pow(sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (theta * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(delta) * cos(phi1)
    if ((theta <= (-8000000.0d0)) .or. (.not. (theta <= 0.41d0))) then
        tmp = lambda1 + atan2((sin(theta) * t_1), (cos(delta) - (sin(phi1) ** 2.0d0)))
    else
        tmp = lambda1 + atan2((sin(delta) * (theta * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + t_1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.sin(delta) * Math.cos(phi1);
	double tmp;
	if ((theta <= -8000000.0) || !(theta <= 0.41)) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * t_1), (Math.cos(delta) - Math.pow(Math.sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (theta * Math.cos(phi1))), (Math.cos(delta) - (Math.sin(phi1) * ((Math.cos(delta) * Math.sin(phi1)) + t_1))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * math.cos(phi1)
	tmp = 0
	if (theta <= -8000000.0) or not (theta <= 0.41):
		tmp = lambda1 + math.atan2((math.sin(theta) * t_1), (math.cos(delta) - math.pow(math.sin(phi1), 2.0)))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (theta * math.cos(phi1))), (math.cos(delta) - (math.sin(phi1) * ((math.cos(delta) * math.sin(phi1)) + t_1))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	tmp = 0.0
	if ((theta <= -8000000.0) || !(theta <= 0.41))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * t_1), Float64(cos(delta) - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(theta * cos(phi1))), Float64(cos(delta) - Float64(sin(phi1) * Float64(Float64(cos(delta) * sin(phi1)) + t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * cos(phi1);
	tmp = 0.0;
	if ((theta <= -8000000.0) || ~((theta <= 0.41)))
		tmp = lambda1 + atan2((sin(theta) * t_1), (cos(delta) - (sin(phi1) ^ 2.0)));
	else
		tmp = lambda1 + atan2((sin(delta) * (theta * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[theta, -8000000.0], N[Not[LessEqual[theta, 0.41]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $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. Split input into 2 regimes

2. if theta < -8e6 or 0.409999999999999976 < theta

   1. Initial program 99.6%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   2. 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. cancel-sign-sub-inv 99.6%

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

      3. cancel-sign-sub 99.6%

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

      4. remove-double-neg 99.6%

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

      5. fma-def 99.6%

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

      6. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in delta around 0 88.7%

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

   if -8e6 < theta < 0.409999999999999976

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. cancel-sign-sub 99.9%

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

      4. remove-double-neg 99.9%

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

      5. fma-def 99.9%

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

      6. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in theta around 0 99.3%

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

   5. Taylor expanded in theta around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -8000000 \lor \neg \left(theta \leq 0.41\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \cos \phi_1\right)}\\ \end{array} \]

Alternative 7: 92.3% 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* 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. cancel-sign-sub 99.8%

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

   4. remove-double-neg 99.8%

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

   5. fma-def 99.7%

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

   6. associate-*l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in delta around 0 90.8%

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

5. Final simplification 90.8%

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

Alternative 8: 91.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ t_2 := \sin delta \cdot t_1\\ \mathbf{if}\;delta \leq -0.045:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (cos phi1))) (t_2 (* (sin delta) t_1)))
   (if (<= delta -0.045)
     (+ lambda1 (atan2 (* (sin delta) (log1p (expm1 t_1))) (cos delta)))
     (if (<= delta 5e-21)
       (+ lambda1 (atan2 t_2 (pow (cos phi1) 2.0)))
       (+ lambda1 (atan2 t_2 (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * cos(phi1);
	double t_2 = sin(delta) * t_1;
	double tmp;
	if (delta <= -0.045) {
		tmp = lambda1 + atan2((sin(delta) * log1p(expm1(t_1))), cos(delta));
	} else if (delta <= 5e-21) {
		tmp = lambda1 + atan2(t_2, pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2(t_2, cos(delta));
	}
	return tmp;
}

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.sin(theta) * Math.cos(phi1);
	double t_2 = Math.sin(delta) * t_1;
	double tmp;
	if (delta <= -0.045) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.log1p(Math.expm1(t_1))), Math.cos(delta));
	} else if (delta <= 5e-21) {
		tmp = lambda1 + Math.atan2(t_2, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2(t_2, Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(theta) * math.cos(phi1)
	t_2 = math.sin(delta) * t_1
	tmp = 0
	if delta <= -0.045:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.log1p(math.expm1(t_1))), math.cos(delta))
	elif delta <= 5e-21:
		tmp = lambda1 + math.atan2(t_2, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2(t_2, math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * cos(phi1))
	t_2 = Float64(sin(delta) * t_1)
	tmp = 0.0
	if (delta <= -0.045)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * log1p(expm1(t_1))), cos(delta)));
	elseif (delta <= 5e-21)
		tmp = Float64(lambda1 + atan(t_2, (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(t_2, cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[delta], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[delta, -0.045], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 5e-21], N[(lambda1 + N[ArcTan[t$95$2 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$2 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -0.044999999999999998

   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. *-commutative 99.7%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 84.0%

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

      1. log1p-expm1-u 84.0%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin theta \cdot \cos \phi_1\right)\right)}}{\cos delta} \]

      2. *-commutative 84.0%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \phi_1 \cdot \sin theta}\right)\right)}{\cos delta} \]

   6. Applied egg-rr 84.0%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin theta\right)\right)}}{\cos delta} \]

   if -0.044999999999999998 < delta < 4.99999999999999973e-21

   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. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in delta around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. unpow2 99.7%

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

      4. 1-sub-sin 99.9%

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

      5. unpow2 99.9%

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

   8. Simplified 99.9%

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

   if 4.99999999999999973e-21 < delta

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 86.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.045:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin theta \cdot \cos \phi_1\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 9: 91.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -50 \lor \neg \left(delta \leq 6.3 \cdot 10^{-18}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{{\cos \phi_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (* (sin theta) (cos phi1)))))
   (if (or (<= delta -50.0) (not (<= delta 6.3e-18)))
     (+ lambda1 (atan2 t_1 (cos delta)))
     (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * (sin(theta) * cos(phi1));
	double tmp;
	if ((delta <= -50.0) || !(delta <= 6.3e-18)) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, 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) :: t_1
    real(8) :: tmp
    t_1 = sin(delta) * (sin(theta) * cos(phi1))
    if ((delta <= (-50.0d0)) .or. (.not. (delta <= 6.3d-18))) then
        tmp = lambda1 + atan2(t_1, cos(delta))
    else
        tmp = lambda1 + atan2(t_1, (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 t_1 = Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1));
	double tmp;
	if ((delta <= -50.0) || !(delta <= 6.3e-18)) {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * (math.sin(theta) * math.cos(phi1))
	tmp = 0
	if (delta <= -50.0) or not (delta <= 6.3e-18):
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * Float64(sin(theta) * cos(phi1)))
	tmp = 0.0
	if ((delta <= -50.0) || !(delta <= 6.3e-18))
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * (sin(theta) * cos(phi1));
	tmp = 0.0;
	if ((delta <= -50.0) || ~((delta <= 6.3e-18)))
		tmp = lambda1 + atan2(t_1, cos(delta));
	else
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[delta, -50.0], N[Not[LessEqual[delta, 6.3e-18]], $MachinePrecision]], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if delta < -50 or 6.3000000000000004e-18 < delta

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 85.5%

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

   if -50 < delta < 6.3000000000000004e-18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in delta around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. unpow2 99.7%

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

      4. 1-sub-sin 99.9%

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

      5. unpow2 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -50 \lor \neg \left(delta \leq 6.3 \cdot 10^{-18}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \end{array} \]

Alternative 10: 86.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -60000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left|\sin theta\right|}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -60000000.0)
   (+ lambda1 (atan2 (* (sin delta) (fabs (sin theta))) (cos delta)))
   (+ lambda1 (atan2 (* (sin delta) (sin theta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -60000000.0) {
		tmp = lambda1 + atan2((sin(delta) * fabs(sin(theta))), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), 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 <= (-60000000.0d0)) then
        tmp = lambda1 + atan2((sin(delta) * abs(sin(theta))), cos(delta))
    else
        tmp = lambda1 + atan2((sin(delta) * sin(theta)), 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 <= -60000000.0) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.abs(Math.sin(theta))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -60000000.0:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.fabs(math.sin(theta))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -60000000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * abs(sin(theta))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -60000000.0)
		tmp = lambda1 + atan2((sin(delta) * abs(sin(theta))), cos(delta));
	else
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -60000000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Abs[N[Sin[theta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6e7

   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. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in phi1 around 0 84.3%

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

   5. Taylor expanded in phi1 around 0 76.7%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-unprod 82.5%

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

      3. pow2 82.5%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sqrt{\color{blue}{{\sin theta}^{2}}}}{\cos delta} \]

   7. Applied egg-rr 82.5%

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

      1. unpow2 82.5%

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

      2. rem-sqrt-square 82.5%

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

   9. Simplified 82.5%

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

   if -6e7 < phi1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 90.0%

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

   5. Taylor expanded in phi1 around 0 87.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -60000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left|\sin theta\right|}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \end{array} \]

Alternative 11: 88.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in phi1 around 0 88.4%

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

5. Final simplification 88.4%

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

Alternative 12: 86.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in phi1 around 0 88.4%

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

5. Taylor expanded in phi1 around 0 84.7%

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

6. Final simplification 84.7%

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

Alternative 13: 80.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -42000000000 \lor \neg \left(delta \leq 4 \cdot 10^{+18}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -42000000000.0) (not (<= delta 4e+18)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
   (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -42000000000.0) || !(delta <= 4e+18)) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = lambda1 + atan2((delta * sin(theta)), 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 <= (-42000000000.0d0)) .or. (.not. (delta <= 4d+18))) then
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    else
        tmp = lambda1 + atan2((delta * sin(theta)), 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 <= -42000000000.0) || !(delta <= 4e+18)) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -42000000000.0) or not (delta <= 4e+18):
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -42000000000.0) || !(delta <= 4e+18))
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -42000000000.0) || ~((delta <= 4e+18)))
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	else
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -42000000000.0], N[Not[LessEqual[delta, 4e+18]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.2e10 or 4e18 < 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. *-commutative 99.7%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 85.9%

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

   5. Taylor expanded in phi1 around 0 79.9%

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

   6. Taylor expanded in theta around 0 68.5%

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

   if -4.2e10 < delta < 4e18

   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. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in phi1 around 0 90.7%

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

   5. Taylor expanded in phi1 around 0 89.1%

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

   6. Taylor expanded in delta around 0 89.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -42000000000 \lor \neg \left(delta \leq 4 \cdot 10^{+18}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \end{array} \]

Alternative 14: 73.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in phi1 around 0 88.4%

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

5. Taylor expanded in phi1 around 0 84.7%

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

6. Taylor expanded in theta around 0 74.6%

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

7. Final simplification 74.6%

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

Alternative 15: 72.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{0}{\cos delta} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ lambda1 (atan2 0.0 (cos delta))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(0.0, 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(0.0d0, cos(delta))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(0.0, Math.cos(delta));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(0.0, math.cos(delta))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(0.0, cos(delta)))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(0.0, cos(delta));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[0.0 / 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. *-commutative 99.7%

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in phi1 around 0 88.4%

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

   1. associate-*r* 88.4%

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

   2. *-commutative 88.4%

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

   3. associate-*r* 88.4%

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

   4. add-log-exp 80.8%

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

   5. *-commutative 80.8%

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

   6. exp-prod 80.8%

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

   7. *-commutative 80.8%

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

6. Applied egg-rr 80.8%

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

   1. add-cube-cbrt 80.8%

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

   2. pow3 80.8%

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

8. Applied egg-rr 80.8%

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

9. Taylor expanded in delta around 0 73.6%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

10. Final simplification 73.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{\cos delta} \]

Alternative 16: 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. *-commutative 99.7%

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

   2. associate-*l* 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in phi1 around 0 88.4%

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

5. Taylor expanded in lambda1 around inf 69.4%

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

6. Final simplification 69.4%

   \[\leadsto \lambda_1 \]

Reproduce

herbie shell --seed 2023229 
(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))))))))))