Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 19.0s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

   1. sin-asin 99.7%

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

   2. fma-undefine 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

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))
      (* (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) * ((cos(delta) * sin(phi1)) + (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) * ((cos(delta) * sin(phi1)) + (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.cos(delta) * Math.sin(phi1)) + (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.cos(delta) * math.sin(phi1)) + (math.cos(phi1) * (math.sin(delta) * 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(cos(phi1) * Float64(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) * ((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * 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[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $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* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in delta around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in delta around inf 99.7%

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

6. Taylor expanded in theta around inf 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 94.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in theta around 0 94.8%

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

6. Final simplification 94.8%

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

Alternative 5: 94.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (-
    (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((cos(phi1) * (sin(theta) * sin(delta))), (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((cos(phi1) * (sin(theta) * sin(delta))), (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.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), (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.cos(phi1) * (math.sin(theta) * math.sin(delta))), (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(cos(phi1) * Float64(sin(theta) * sin(delta))), 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((cos(phi1) * (sin(theta) * sin(delta))), (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[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[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%

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

3. Simplified 99.7%

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

   1. sin-asin 99.7%

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

   2. fma-undefine 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in theta around inf 99.7%

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

8. Taylor expanded in theta around 0 94.8%

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

9. Final simplification 94.8%

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

Alternative 6: 94.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    t_1 = sin(delta) * cos(phi1)
    code = lambda1 + atan2((sin(theta) * t_1), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in theta around 0 94.8%

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

6. Final simplification 94.8%

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

Alternative 7: 92.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in delta around 0 91.8%

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

Alternative 8: 92.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in delta around 0 91.8%

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

Alternative 9: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.00019:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \frac{1}{\frac{1}{t\_1}}}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (if (<= delta -1.6e-5)
     (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
     (if (<= delta 0.00019)
       (+ lambda1 (atan2 (* (sin theta) t_1) (pow (cos phi1) 2.0)))
       (+ lambda1 (atan2 (* (sin theta) (/ 1.0 (/ 1.0 t_1))) (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * cos(phi1);
	double tmp;
	if (delta <= -1.6e-5) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else if (delta <= 0.00019) {
		tmp = lambda1 + atan2((sin(theta) * t_1), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((sin(theta) * (1.0 / (1.0 / t_1))), 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) :: t_1
    real(8) :: tmp
    t_1 = sin(delta) * cos(phi1)
    if (delta <= (-1.6d-5)) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else if (delta <= 0.00019d0) then
        tmp = lambda1 + atan2((sin(theta) * t_1), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((sin(theta) * (1.0d0 / (1.0d0 / t_1))), cos(delta))
    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 (delta <= -1.6e-5) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else if (delta <= 0.00019) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * t_1), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (1.0 / (1.0 / t_1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * math.cos(phi1)
	tmp = 0
	if delta <= -1.6e-5:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	elif delta <= 0.00019:
		tmp = lambda1 + math.atan2((math.sin(theta) * t_1), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * (1.0 / (1.0 / t_1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	tmp = 0.0
	if (delta <= -1.6e-5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	elseif (delta <= 0.00019)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * t_1), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(1.0 / Float64(1.0 / t_1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * cos(phi1);
	tmp = 0.0;
	if (delta <= -1.6e-5)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	elseif (delta <= 0.00019)
		tmp = lambda1 + atan2((sin(theta) * t_1), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((sin(theta) * (1.0 / (1.0 / t_1))), cos(delta));
	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[LessEqual[delta, -1.6e-5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 0.00019], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(1.0 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.59999999999999993e-5

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. fma-define 99.8%

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

      4. cos-neg 99.8%

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

      5. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 80.1%

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

   6. Taylor expanded in theta around 0 80.1%

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

   if -1.59999999999999993e-5 < delta < 1.9000000000000001e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. sin-asin 99.7%

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

      2. fma-undefine 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in delta around 0 99.4%

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

      1. neg-mul-1 99.4%

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

      2. sub-neg 99.4%

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

      3. unpow2 99.4%

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

      4. 1-sub-sin 99.6%

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

      5. unpow2 99.6%

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

   9. Simplified 99.6%

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

   if 1.9000000000000001e-4 < delta

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 83.2%

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

      1. sin-cos-mult 81.9%

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

      2. clear-num 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. *-un-lft-identity 81.9%

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

      2. clear-num 81.9%

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

      3. sin-cos-mult 83.2%

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

      4. *-commutative 83.2%

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.00019:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \frac{1}{\frac{1}{\sin delta \cdot \cos \phi_1}}}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -0.0105:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \frac{1}{\frac{1}{\sin delta \cdot \cos \phi_1}}}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta)))))
   (if (<= delta -0.0105)
     (+ lambda1 (atan2 t_1 (cos delta)))
     (if (<= delta 8.8e-5)
       (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0)))
       (+
        lambda1
        (atan2
         (* (sin theta) (/ 1.0 (/ 1.0 (* (sin delta) (cos phi1)))))
         (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double tmp;
	if (delta <= -0.0105) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else if (delta <= 8.8e-5) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((sin(theta) * (1.0 / (1.0 / (sin(delta) * cos(phi1))))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(theta) * sin(delta))
    if (delta <= (-0.0105d0)) then
        tmp = lambda1 + atan2(t_1, cos(delta))
    else if (delta <= 8.8d-5) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((sin(theta) * (1.0d0 / (1.0d0 / (sin(delta) * cos(phi1))))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta));
	double tmp;
	if (delta <= -0.0105) {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	} else if (delta <= 8.8e-5) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (1.0 / (1.0 / (Math.sin(delta) * Math.cos(phi1))))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(theta) * math.sin(delta))
	tmp = 0
	if delta <= -0.0105:
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	elif delta <= 8.8e-5:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * (1.0 / (1.0 / (math.sin(delta) * math.cos(phi1))))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	tmp = 0.0
	if (delta <= -0.0105)
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	elseif (delta <= 8.8e-5)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(1.0 / Float64(1.0 / Float64(sin(delta) * cos(phi1))))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(theta) * sin(delta));
	tmp = 0.0;
	if (delta <= -0.0105)
		tmp = lambda1 + atan2(t_1, cos(delta));
	elseif (delta <= 8.8e-5)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((sin(theta) * (1.0 / (1.0 / (sin(delta) * cos(phi1))))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.0105], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 8.8e-5], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(1.0 / N[(1.0 / N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -0.0105000000000000007

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. fma-define 99.8%

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

      4. cos-neg 99.8%

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

      5. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 80.1%

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

   6. Taylor expanded in theta around 0 80.1%

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

   if -0.0105000000000000007 < delta < 8.7999999999999998e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. sin-asin 99.7%

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

      2. fma-undefine 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in theta around inf 99.7%

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

   8. Taylor expanded in delta around 0 99.4%

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

      1. neg-mul-1 99.4%

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

      2. sub-neg 99.4%

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

      3. unpow2 99.4%

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

      4. 1-sub-sin 99.6%

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

      5. unpow2 99.6%

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

   10. Simplified 99.6%

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

   if 8.7999999999999998e-5 < delta

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 83.2%

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

      1. sin-cos-mult 81.9%

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

      2. clear-num 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. *-un-lft-identity 81.9%

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

      2. clear-num 81.9%

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

      3. sin-cos-mult 83.2%

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

      4. *-commutative 83.2%

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.0105:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \frac{1}{\frac{1}{\sin delta \cdot \cos \phi_1}}}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.000115:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \frac{1}{\frac{1}{\sin delta \cdot \cos \phi_1}}}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -0.000115)
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
   (if (<= delta 5.7e-5)
     (+
      lambda1
      (atan2
       (* (cos phi1) (* (sin theta) delta))
       (- 1.0 (pow (sin phi1) 2.0))))
     (+
      lambda1
      (atan2
       (* (sin theta) (/ 1.0 (/ 1.0 (* (sin delta) (cos phi1)))))
       (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.000115) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else if (delta <= 5.7e-5) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 - pow(sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + atan2((sin(theta) * (1.0 / (1.0 / (sin(delta) * cos(phi1))))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= (-0.000115d0)) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else if (delta <= 5.7d-5) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0d0 - (sin(phi1) ** 2.0d0)))
    else
        tmp = lambda1 + atan2((sin(theta) * (1.0d0 / (1.0d0 / (sin(delta) * cos(phi1))))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.000115) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else if (delta <= 5.7e-5) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (1.0 / (1.0 / (Math.sin(delta) * Math.cos(phi1))))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -0.000115:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	elif delta <= 5.7e-5:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (1.0 - math.pow(math.sin(phi1), 2.0)))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * (1.0 / (1.0 / (math.sin(delta) * math.cos(phi1))))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -0.000115)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	elseif (delta <= 5.7e-5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(1.0 - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(1.0 / Float64(1.0 / Float64(sin(delta) * cos(phi1))))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -0.000115)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	elseif (delta <= 5.7e-5)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 - (sin(phi1) ^ 2.0)));
	else
		tmp = lambda1 + atan2((sin(theta) * (1.0 / (1.0 / (sin(delta) * cos(phi1))))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -0.000115], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 5.7e-5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(1.0 / N[(1.0 / N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.15e-4

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. fma-define 99.8%

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

      4. cos-neg 99.8%

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

      5. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 80.1%

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

   6. Taylor expanded in theta around 0 80.1%

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

   if -1.15e-4 < delta < 5.7000000000000003e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-expm1-u 99.7%

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

      2. log1p-undefine 99.6%

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

      3. sin-asin 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. associate-*r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in delta around 0 99.4%

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

      1. expm1-define 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in delta around 0 99.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}}{\mathsf{fma}\left(\log \left(1 + \mathsf{expm1}\left(\sin \phi_1\right)\right), -\sin \phi_1, \cos delta\right)} \]
   11. Step-by-step derivation

       1. *-commutative 99.4%

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

       2. associate-*l* 99.4%

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

       3. *-commutative 99.4%

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

   12. Simplified 99.4%

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

   13. Taylor expanded in delta around 0 99.4%

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

   if 5.7000000000000003e-5 < delta

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 83.2%

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

      1. sin-cos-mult 81.9%

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

      2. clear-num 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. *-un-lft-identity 81.9%

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

      2. clear-num 81.9%

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

      3. sin-cos-mult 83.2%

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

      4. *-commutative 83.2%

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.000115:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \frac{1}{\frac{1}{\sin delta \cdot \cos \phi_1}}}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.000155 \lor \neg \left(delta \leq 0.00019\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 - {\sin \phi_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -0.000155) (not (<= delta 0.00019)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
   (+
    lambda1
    (atan2
     (* (cos phi1) (* (sin theta) delta))
     (- 1.0 (pow (sin phi1) 2.0))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -0.000155) || !(delta <= 0.00019)) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 - pow(sin(phi1), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-0.000155d0)) .or. (.not. (delta <= 0.00019d0))) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0d0 - (sin(phi1) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -0.000155) || !(delta <= 0.00019)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -0.000155) or not (delta <= 0.00019):
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (1.0 - math.pow(math.sin(phi1), 2.0)))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -0.000155) || !(delta <= 0.00019))
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(1.0 - (sin(phi1) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -0.000155) || ~((delta <= 0.00019)))
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	else
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 - (sin(phi1) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -0.000155], N[Not[LessEqual[delta, 0.00019]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.55e-4 or 1.9000000000000001e-4 < delta

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 81.4%

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

   6. Taylor expanded in theta around 0 81.4%

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

   if -1.55e-4 < delta < 1.9000000000000001e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-expm1-u 99.7%

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

      2. log1p-undefine 99.6%

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

      3. sin-asin 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. associate-*r* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in delta around 0 99.4%

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

      1. expm1-define 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in delta around 0 99.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}}{\mathsf{fma}\left(\log \left(1 + \mathsf{expm1}\left(\sin \phi_1\right)\right), -\sin \phi_1, \cos delta\right)} \]
   11. Step-by-step derivation

       1. *-commutative 99.4%

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

       2. associate-*l* 99.4%

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

       3. *-commutative 99.4%

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

   12. Simplified 99.4%

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

   13. Taylor expanded in delta around 0 99.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.000155 \lor \neg \left(delta \leq 0.00019\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 - {\sin \phi_1}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in phi1 around 0 85.5%

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

6. Taylor expanded in theta around 0 85.5%

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

7. Final simplification 85.5%

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

Alternative 14: 86.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -1.35)
   (+ lambda1 (atan2 (* (sin theta) (* (sin delta) (cos phi1))) 1.0))
   (+ lambda1 (atan2 (* (sin theta) (sin delta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.35) {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), 1.0);
	} else {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-1.35d0)) then
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), 1.0d0)
    else
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.35) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), 1.0);
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -1.35:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), 1.0)
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -1.35)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), 1.0));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -1.35)
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), 1.0);
	else
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -1.35], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.3500000000000001

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in delta around 0 75.7%

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

   6. Taylor expanded in phi1 around 0 72.2%

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

   if -1.3500000000000001 < 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. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 91.1%

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

   6. Taylor expanded in phi1 around 0 89.3%

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

Alternative 15: 86.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in phi1 around 0 85.5%

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

6. Taylor expanded in phi1 around 0 83.1%

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

Alternative 16: 80.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -9.2 \cdot 10^{+77} \lor \neg \left(theta \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -9.2e+77) (not (<= theta 5e-13)))
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -9.2e+77) || !(theta <= 5e-13)) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((theta <= (-9.2d+77)) .or. (.not. (theta <= 5d-13))) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -9.2e+77) || !(theta <= 5e-13)) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (theta <= -9.2e+77) or not (theta <= 5e-13):
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -9.2e+77) || !(theta <= 5e-13))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((theta <= -9.2e+77) || ~((theta <= 5e-13)))
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -9.2e+77], N[Not[LessEqual[theta, 5e-13]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -9.19999999999999979e77 or 4.9999999999999999e-13 < theta

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 81.4%

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

   6. Taylor expanded in phi1 around 0 79.2%

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

   7. Taylor expanded in delta around 0 68.5%

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

   if -9.19999999999999979e77 < theta < 4.9999999999999999e-13

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. fma-define 99.7%

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

      4. cos-neg 99.7%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in phi1 around 0 88.8%

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

   6. Taylor expanded in phi1 around 0 86.1%

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

   7. Taylor expanded in theta around 0 84.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -9.2 \cdot 10^{+77} \lor \neg \left(theta \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in phi1 around 0 85.5%

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

6. Taylor expanded in phi1 around 0 83.1%

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

7. Taylor expanded in delta around 0 72.6%

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

8. Final simplification 72.6%

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

Alternative 18: 69.8% accurate, 1320.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. fma-define 99.7%

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

   4. cos-neg 99.7%

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

   5. 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. Add Preprocessing

5. Taylor expanded in phi1 around 0 85.5%

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

6. Taylor expanded in phi1 around 0 83.1%

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

7. Taylor expanded in lambda1 around inf 69.9%

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

Reproduce

herbie shell --seed 2024090 
(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))))))))))