Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 23.6s

Alternatives: 12

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, 0.9× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin delta) (sin theta)) (cos phi1))
   (fma
    (- (sin phi1))
    (fma
     (log1p (expm1 (cos phi1)))
     (* (sin delta) (cos theta))
     (* (sin phi1) (cos delta)))
    (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(delta) * sin(theta)) * cos(phi1)), fma(-sin(phi1), fma(log1p(expm1(cos(phi1))), (sin(delta) * cos(theta)), (sin(phi1) * cos(delta))), cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(delta) * sin(theta)) * cos(phi1)), fma(Float64(-sin(phi1)), fma(log1p(expm1(cos(phi1))), Float64(sin(delta) * cos(theta)), Float64(sin(phi1) * cos(delta))), cos(delta))))
end

Wolfram

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

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. sin-asin 99.9%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.8%

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

   4. add-cube-cbrt 99.8%

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

   5. pow3 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0 99.9%

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

   1. *-commutative 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-undefine 99.9%

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

   7. distribute-lft-neg-in 99.9%

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

   8. +-commutative 99.9%

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

9. Simplified 99.9%

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

    1. log1p-expm1-u 99.9%

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

11. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. sin-asin 99.9%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.8%

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

   4. add-cube-cbrt 99.8%

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

   5. pow3 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0 99.9%

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

   1. *-commutative 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-undefine 99.9%

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

   7. distribute-lft-neg-in 99.9%

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

   8. +-commutative 99.9%

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

9. Simplified 99.9%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. sin-asin 99.9%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.8%

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

   4. add-cube-cbrt 99.8%

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

   5. pow3 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0 99.9%

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

   1. *-commutative 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-undefine 99.9%

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

   7. distribute-lft-neg-in 99.9%

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

   8. +-commutative 99.9%

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

9. Simplified 99.9%

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

    1. log1p-expm1-u 99.9%

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

11. Applied egg-rr 99.9%

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

    1. log1p-expm1-u 99.9%

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

    2. fma-undefine 99.9%

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

13. Applied egg-rr 99.9%

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

14. Final simplification 99.9%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (-
    (cos delta)
    (*
     (sin phi1)
     (+
      (* (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(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(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(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(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(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(cos(delta) - Float64(sin(phi1) * Float64(Float64(sin(phi1) * cos(delta)) + 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) * ((sin(phi1) * cos(delta)) + (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[Sin[phi1], $MachinePrecision] * N[Cos[delta], $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 99.9%

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

   1. associate-*l* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in delta around inf 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 94.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. sin-asin 99.9%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.8%

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

   4. add-cube-cbrt 99.8%

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

   5. pow3 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0 99.9%

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

   1. *-commutative 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-undefine 99.9%

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

   7. distribute-lft-neg-in 99.9%

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

   8. +-commutative 99.9%

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

9. Simplified 99.9%

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

    1. log1p-expm1-u 99.9%

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

11. Applied egg-rr 99.9%

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

12. Taylor expanded in theta around 0 95.0%

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

13. Final simplification 95.0%

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

Alternative 6: 94.5% accurate, 1.3× speedup

Math

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

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) (+ (* (sin phi1) (cos delta)) 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) * ((sin(phi1) * cos(delta)) + 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) * ((sin(phi1) * cos(delta)) + 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.sin(phi1) * Math.cos(delta)) + 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.sin(phi1) * math.cos(delta)) + 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(sin(phi1) * cos(delta)) + 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) * ((sin(phi1) * cos(delta)) + 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[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in theta around 0 95.0%

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

6. Final simplification 95.0%

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

Alternative 7: 94.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in theta around inf 99.9%

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

6. Taylor expanded in theta around 0 95.0%

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

7. Final simplification 95.0%

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

Alternative 8: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - {\left(\left|\sqrt[3]{\sin \phi_1}\right|\right)}^{6}} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (- (cos delta) (pow (fabs (cbrt (sin phi1))) 6.0)))))

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(fabs(cbrt(sin(phi1))), 6.0)));
}

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.abs(Math.cbrt(Math.sin(phi1))), 6.0)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(cos(delta) - (abs(cbrt(sin(phi1))) ^ 6.0))))
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[Abs[N[Power[N[Sin[phi1], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. sin-asin 99.9%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.8%

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

   4. add-cube-cbrt 99.8%

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

   5. pow3 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in delta around 0 91.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - {\color{blue}{\left(\sqrt[3]{{\sin \phi_1}^{2}}\right)}}^{3}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 91.1%

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

   2. unpow-prod-down 91.1%

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

   3. unpow2 91.1%

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

   4. cbrt-prod 91.1%

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

   5. pow2 91.1%

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

   6. unpow2 91.1%

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

   7. cbrt-prod 91.1%

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

   8. pow2 91.1%

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

9. Applied egg-rr 91.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{{\left(\sqrt{{\left(\sqrt[3]{\sin \phi_1}\right)}^{2}}\right)}^{3} \cdot {\left(\sqrt{{\left(\sqrt[3]{\sin \phi_1}\right)}^{2}}\right)}^{3}}} \]
10. Step-by-step derivation

    1. pow-sqr 91.1%

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

    2. unpow2 91.1%

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

    3. rem-sqrt-square 91.1%

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

    4. metadata-eval 91.1%

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

11. Simplified 91.1%

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

Alternative 9: 92.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in delta around 0 91.1%

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

Alternative 10: 92.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), (math.cos(delta) + ((math.cos((phi1 * 2.0)) / 2.0) - 0.5)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

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

   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 99.9%

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

   4. cos-neg 99.9%

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

   5. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. sin-asin 99.9%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.8%

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

   4. add-cube-cbrt 99.8%

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

   5. pow3 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in delta around 0 91.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - {\color{blue}{\left(\sqrt[3]{{\sin \phi_1}^{2}}\right)}}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 91.1%

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

   2. unpow2 91.1%

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

   3. sin-mult 90.7%

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

9. Applied egg-rr 90.7%

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

    1. div-sub 90.7%

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

    2. +-inverses 90.7%

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

    3. cos-0 90.7%

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

    4. metadata-eval 90.7%

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

    5. count-2 90.7%

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

    6. *-commutative 90.7%

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

11. Simplified 90.7%

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

12. Final simplification 90.7%

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

Alternative 11: 89.1% accurate, 2.6× speedup

Math

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

FPCore

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(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[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in phi1 around 0 90.0%

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

Alternative 12: 70.6% 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 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in lambda1 around inf 76.4%

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

Reproduce

herbie shell --seed 2024157 
(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))))))))))