Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 23.1s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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 inf 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. sin-asin 99.7%

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

7. Applied egg-rr 99.7%

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

   1. fma-undefine 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin delta) (sin theta)))
   (-
    (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(delta) * sin(theta))), (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(delta) * sin(theta))), (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(delta) * Math.sin(theta))), (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(delta) * math.sin(theta))), (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(delta) * sin(theta))), 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(delta) * sin(theta))), (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[delta], $MachinePrecision] * N[Sin[theta], $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. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in 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 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 94.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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 inf 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. sin-asin 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in theta around 0 95.7%

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

9. Final simplification 95.7%

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

Alternative 4: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 4.6 \cdot 10^{+106}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 4.6e+106)
   (+
    lambda1
    (atan2
     (* (sin theta) (* (cos phi1) (sin delta)))
     (- (cos delta) (pow (sin phi1) 2.0))))
   (+
    lambda1
    (atan2
     (* (sin delta) (sin theta))
     (-
      (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) {
	double tmp;
	if (delta <= 4.6e+106) {
		tmp = lambda1 + atan2((sin(theta) * (cos(phi1) * sin(delta))), (cos(delta) - pow(sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * cos(theta)))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= 4.6d+106) then
        tmp = lambda1 + atan2((sin(theta) * (cos(phi1) * sin(delta))), (cos(delta) - (sin(phi1) ** 2.0d0)))
    else
        tmp = lambda1 + atan2((sin(delta) * sin(theta)), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * cos(theta)))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 4.6e+106) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.cos(phi1) * Math.sin(delta))), (Math.cos(delta) - Math.pow(Math.sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), (Math.cos(delta) - (Math.sin(phi1) * ((Math.cos(delta) * Math.sin(phi1)) + (Math.cos(phi1) * (Math.sin(delta) * Math.cos(theta)))))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= 4.6e+106:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.cos(phi1) * math.sin(delta))), (math.cos(delta) - math.pow(math.sin(phi1), 2.0)))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), (math.cos(delta) - (math.sin(phi1) * ((math.cos(delta) * math.sin(phi1)) + (math.cos(phi1) * (math.sin(delta) * math.cos(theta)))))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 4.6e+106)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(cos(phi1) * sin(delta))), Float64(cos(delta) - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), Float64(cos(delta) - Float64(sin(phi1) * Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(phi1) * Float64(sin(delta) * cos(theta))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= 4.6e+106)
		tmp = lambda1 + atan2((sin(theta) * (cos(phi1) * sin(delta))), (cos(delta) - (sin(phi1) ^ 2.0)));
	else
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * cos(theta)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 4.6e+106], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $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. Split input into 2 regimes

2. if delta < 4.6000000000000004e106

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in delta around 0 95.8%

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

   if 4.6000000000000004e106 < delta

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in delta around inf 99.8%

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

   6. Taylor expanded in phi1 around 0 85.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq 4.6 \cdot 10^{+106}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (sin delta))))
   (+
    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 = cos(phi1) * sin(delta);
	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 = cos(phi1) * sin(delta)
    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.cos(phi1) * Math.sin(delta);
	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.cos(phi1) * math.sin(delta)
	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(cos(phi1) * sin(delta))
	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 = cos(phi1) * sin(delta);
	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[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in theta around 0 95.7%

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

6. Final simplification 95.7%

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

Alternative 6: 92.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in delta around 0 92.7%

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

6. Final simplification 92.7%

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

Alternative 7: 90.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ \mathbf{if}\;\phi_1 \leq -1.8 \lor \neg \left(\phi_1 \leq 0.1\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta))))
   (if (or (<= phi1 -1.8) (not (<= phi1 0.1)))
     (+ lambda1 (atan2 (* (cos phi1) t_1) (pow (cos phi1) 2.0)))
     (+ lambda1 (atan2 t_1 (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double tmp;
	if ((phi1 <= -1.8) || !(phi1 <= 0.1)) {
		tmp = lambda1 + atan2((cos(phi1) * t_1), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2(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) * sin(theta)
    if ((phi1 <= (-1.8d0)) .or. (.not. (phi1 <= 0.1d0))) then
        tmp = lambda1 + atan2((cos(phi1) * t_1), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2(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.sin(theta);
	double tmp;
	if ((phi1 <= -1.8) || !(phi1 <= 0.1)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * t_1), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * math.sin(theta)
	tmp = 0
	if (phi1 <= -1.8) or not (phi1 <= 0.1):
		tmp = lambda1 + math.atan2((math.cos(phi1) * t_1), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	tmp = 0.0
	if ((phi1 <= -1.8) || !(phi1 <= 0.1))
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * t_1), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * sin(theta);
	tmp = 0.0;
	if ((phi1 <= -1.8) || ~((phi1 <= 0.1)))
		tmp = lambda1 + atan2((cos(phi1) * t_1), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2(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[Sin[theta], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1.8], N[Not[LessEqual[phi1, 0.1]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.80000000000000004 or 0.10000000000000001 < phi1

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in theta around inf 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. sin-asin 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in delta around 0 84.1%

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

      1. mul-1-neg 84.1%

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

      2. unsub-neg 84.1%

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

      3. unpow2 84.1%

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

      4. 1-sub-sin 84.2%

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

      5. unpow2 84.2%

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

   10. Simplified 84.2%

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

   if -1.80000000000000004 < phi1 < 0.10000000000000001

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. cos-neg 99.9%

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

      3. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi1 around 0 99.4%

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

   6. Taylor expanded in phi1 around 0 99.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8 \lor \neg \left(\phi_1 \leq 0.1\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta - \phi_1 \cdot \sin delta}\\ \mathbf{elif}\;delta \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (* (cos phi1) (sin delta)))))
   (if (<= delta -5.7e-6)
     (+ lambda1 (atan2 t_1 (- (cos delta) (* phi1 (sin delta)))))
     (if (<= delta 8.7e-16)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin delta) (sin theta)))
         (pow (cos phi1) 2.0)))
       (+ lambda1 (atan2 t_1 (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * (cos(phi1) * sin(delta));
	double tmp;
	if (delta <= -5.7e-6) {
		tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * sin(delta))));
	} else if (delta <= 8.7e-16) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2(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(theta) * (cos(phi1) * sin(delta))
    if (delta <= (-5.7d-6)) then
        tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * sin(delta))))
    else if (delta <= 8.7d-16) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2(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(theta) * (Math.cos(phi1) * Math.sin(delta));
	double tmp;
	if (delta <= -5.7e-6) {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(delta) - (phi1 * Math.sin(delta))));
	} else if (delta <= 8.7e-16) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(theta) * (math.cos(phi1) * math.sin(delta))
	tmp = 0
	if delta <= -5.7e-6:
		tmp = lambda1 + math.atan2(t_1, (math.cos(delta) - (phi1 * math.sin(delta))))
	elif delta <= 8.7e-16:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * Float64(cos(phi1) * sin(delta)))
	tmp = 0.0
	if (delta <= -5.7e-6)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(delta) - Float64(phi1 * sin(delta)))));
	elseif (delta <= 8.7e-16)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(theta) * (cos(phi1) * sin(delta));
	tmp = 0.0;
	if (delta <= -5.7e-6)
		tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * sin(delta))));
	elseif (delta <= 8.7e-16)
		tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2(t_1, cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -5.7e-6], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 8.7e-16], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -5.6999999999999996e-6

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

      2. cos-neg 99.5%

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

      3. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in theta around 0 95.8%

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

   6. Taylor expanded in phi1 around 0 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

   8. Simplified 88.3%

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

   if -5.6999999999999996e-6 < delta < 8.70000000000000036e-16

   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 inf 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. sin-asin 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in delta around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. unpow2 99.7%

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

      4. 1-sub-sin 99.9%

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

      5. unpow2 99.9%

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

   10. Simplified 99.9%

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

   if 8.70000000000000036e-16 < delta

   1. Initial program 99.8%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 78.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\cos delta - \phi_1 \cdot \sin delta}\\ \mathbf{elif}\;delta \leq 8.7 \cdot 10^{-16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

5. Taylor expanded in phi1 around 0 87.9%

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

6. Final simplification 87.9%

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

Alternative 10: 86.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in phi1 around 0 87.9%

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

6. Taylor expanded in phi1 around 0 87.1%

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

7. Final simplification 87.1%

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

Alternative 11: 80.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -1.7 \cdot 10^{-20} \lor \neg \left(delta \leq 7.5 \cdot 10^{-12}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -1.7e-20) (not (<= delta 7.5e-12)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
   (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -1.7e-20) || !(delta <= 7.5e-12)) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-1.7d-20)) .or. (.not. (delta <= 7.5d-12))) then
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    else
        tmp = lambda1 + atan2((delta * sin(theta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -1.7e-20) || !(delta <= 7.5e-12)) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -1.7e-20) or not (delta <= 7.5e-12):
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -1.7e-20) || !(delta <= 7.5e-12))
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -1.7e-20) || ~((delta <= 7.5e-12)))
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	else
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -1.7e-20], N[Not[LessEqual[delta, 7.5e-12]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.6999999999999999e-20 or 7.5e-12 < 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* 99.6%

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

      2. cos-neg 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in phi1 around 0 82.7%

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

      1. add-cbrt-cube 82.6%

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

      2. pow3 82.6%

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

      3. *-commutative 82.6%

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

   7. Applied egg-rr 82.6%

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

   8. Taylor expanded in phi1 around 0 81.3%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sqrt[3]{\color{blue}{{\sin delta}^{3} \cdot {\sin theta}^{3}}}}{\cos delta} \]

   9. Taylor expanded in theta around 0 70.3%

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

   if -1.6999999999999999e-20 < delta < 7.5e-12

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0 93.0%

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

      1. add-cbrt-cube 88.2%

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

      2. pow3 88.2%

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

      3. *-commutative 88.2%

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

   7. Applied egg-rr 88.2%

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

   8. Taylor expanded in phi1 around 0 88.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sqrt[3]{\color{blue}{{\sin delta}^{3} \cdot {\sin theta}^{3}}}}{\cos delta} \]

   9. Taylor expanded in delta around 0 92.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.7 \cdot 10^{-20} \lor \neg \left(delta \leq 7.5 \cdot 10^{-12}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 1.6e+103)
   (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))
   lambda1))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 1.6e+103) {
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= 1.6d+103) then
        tmp = lambda1 + atan2((delta * sin(theta)), cos(delta))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 1.6e+103) {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= 1.6e+103:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 1.6e+103)
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= 1.6e+103)
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 1.6e+103], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]

Derivation

1. Split input into 2 regimes

2. if delta < 1.59999999999999996e103

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0 90.3%

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

      1. add-cbrt-cube 87.4%

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

      2. pow3 87.4%

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

      3. *-commutative 87.4%

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

   7. Applied egg-rr 87.4%

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

   8. Taylor expanded in phi1 around 0 86.6%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sqrt[3]{\color{blue}{{\sin delta}^{3} \cdot {\sin theta}^{3}}}}{\cos delta} \]

   9. Taylor expanded in delta around 0 82.2%

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

   if 1.59999999999999996e103 < delta

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 72.6%

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

   6. Taylor expanded in lambda1 around inf 53.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.4% accurate, 1320.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in phi1 around 0 87.9%

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

6. Taylor expanded in lambda1 around inf 71.4%

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

7. Final simplification 71.4%

   \[\leadsto \lambda_1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))))))))