Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 18.4s

Alternatives: 12

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. lift-+.f64 N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

8. Applied egg-rr 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. sqr-cos-a N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 99.9

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.8%

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

Alternative 2: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-135}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 + \phi_1\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta - \phi_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2
         (atan2
          t_1
          (-
           (cos delta)
           (*
            (sin phi1)
            (+
             (* (sin phi1) (cos delta))
             (* (* (cos phi1) (sin delta)) (cos theta))))))))
   (if (<= t_2 -2e-135)
     (+ lambda1 (atan2 t_1 (cos delta)))
     (if (<= t_2 5e-38)
       (+ lambda1 (atan2 t_1 (fma 0.5 (cos (+ phi1 phi1)) 0.5)))
       (+ lambda1 (atan2 t_1 (- (cos delta) (* phi1 phi1))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = atan2(t_1, (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))));
	double tmp;
	if (t_2 <= -2e-135) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else if (t_2 <= 5e-38) {
		tmp = lambda1 + atan2(t_1, fma(0.5, cos((phi1 + phi1)), 0.5));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = atan(t_1, Float64(cos(delta) - Float64(sin(phi1) * Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))
	tmp = 0.0
	if (t_2 <= -2e-135)
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	elseif (t_2 <= 5e-38)
		tmp = Float64(lambda1 + atan(t_1, fma(0.5, cos(Float64(phi1 + phi1)), 0.5)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(delta) - Float64(phi1 * phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * 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]}, If[LessEqual[t$95$2, -2e-135], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-38], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -2.0000000000000001e-135

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.7

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

   5. Simplified 84.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 84.7

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

   7. Applied egg-rr 84.7%

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

   if -2.0000000000000001e-135 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 5.00000000000000033e-38

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.6

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

   8. Simplified 99.6%

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

   if 5.00000000000000033e-38 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 92.8

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

   5. Simplified 92.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.6

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

   8. Simplified 94.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq -2 \cdot 10^{-135}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 + \phi_1\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ \mathbf{if}\;t\_2 \leq -1.45:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{0}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2
         (atan2
          (* (cos phi1) t_1)
          (-
           (cos delta)
           (*
            (sin phi1)
            (+
             (* (sin phi1) (cos delta))
             (* (* (cos phi1) (sin delta)) (cos theta))))))))
   (if (<= t_2 -1.45)
     (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
     (if (<= t_2 1.0)
       (+ lambda1 (atan2 t_1 1.0))
       (+ lambda1 (atan2 0.0 (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))));
	double tmp;
	if (t_2 <= -1.45) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else if (t_2 <= 1.0) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else {
		tmp = lambda1 + atan2(0.0, cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(delta) * sin(theta)
    t_2 = atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))
    if (t_2 <= (-1.45d0)) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else if (t_2 <= 1.0d0) then
        tmp = lambda1 + atan2(t_1, 1.0d0)
    else
        tmp = lambda1 + atan2(0.0d0, cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.sin(delta) * Math.sin(theta);
	double t_2 = Math.atan2((Math.cos(phi1) * t_1), (Math.cos(delta) - (Math.sin(phi1) * ((Math.sin(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))));
	double tmp;
	if (t_2 <= -1.45) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else if (t_2 <= 1.0) {
		tmp = lambda1 + Math.atan2(t_1, 1.0);
	} else {
		tmp = lambda1 + Math.atan2(0.0, Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * math.sin(theta)
	t_2 = math.atan2((math.cos(phi1) * t_1), (math.cos(delta) - (math.sin(phi1) * ((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))
	tmp = 0
	if t_2 <= -1.45:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	elif t_2 <= 1.0:
		tmp = lambda1 + math.atan2(t_1, 1.0)
	else:
		tmp = lambda1 + math.atan2(0.0, math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))
	tmp = 0.0
	if (t_2 <= -1.45)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	elseif (t_2 <= 1.0)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	else
		tmp = Float64(lambda1 + atan(0.0, cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * sin(theta);
	t_2 = atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))));
	tmp = 0.0;
	if (t_2 <= -1.45)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	elseif (t_2 <= 1.0)
		tmp = lambda1 + atan2(t_1, 1.0);
	else
		tmp = lambda1 + atan2(0.0, cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[phi1], $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] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1.45], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[0.0 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -1.44999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 85.4

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

   5. Simplified 85.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 80.1

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

   8. Simplified 80.1%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 74.0

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

   11. Simplified 74.0%

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

   if -1.44999999999999996 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 91.2

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

   5. Simplified 91.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 90.3

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

   8. Simplified 90.3%

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

   9. Taylor expanded in delta around 0

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

   11. Simplified 86.1%

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

   if 1 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.4

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

   5. Simplified 90.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 84.4

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

   8. Simplified 84.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 88.2

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

   10. Applied egg-rr 88.2%

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

   11. Taylor expanded in theta around 0

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

       1. cos-neg N/A

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

       2. +-inverses 88.2

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0} \cdot 0.5}{\cos delta} \]

   13. Simplified 88.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0} \cdot 0.5}{\cos delta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq -1.45:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{0}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{0}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<=
      (+
       lambda1
       (atan2
        (* (cos phi1) (* (sin delta) (sin theta)))
        (-
         (cos delta)
         (*
          (sin phi1)
          (+
           (* (sin phi1) (cos delta))
           (* (* (cos phi1) (sin delta)) (cos theta)))))))
      2e-84)
   (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))
   (+ lambda1 (atan2 0.0 (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))) <= 2e-84) {
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	} else {
		tmp = lambda1 + atan2(0.0, cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))) <= 2d-84) then
        tmp = lambda1 + atan2((delta * sin(theta)), cos(delta))
    else
        tmp = lambda1 + atan2(0.0d0, cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), (Math.cos(delta) - (Math.sin(phi1) * ((Math.sin(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))))) <= 2e-84) {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(0.0, Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (lambda1 + math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), (math.cos(delta) - (math.sin(phi1) * ((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))) <= 2e-84:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(0.0, math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(cos(delta) - Float64(sin(phi1) * Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))) <= 2e-84)
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(0.0, cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (sin(phi1) * ((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))) <= 2e-84)
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	else
		tmp = lambda1 + atan2(0.0, cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[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], 2e-84], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[0.0 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 2.0000000000000001e-84

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 88.5

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

   5. Simplified 88.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 86.6

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

   8. Simplified 86.6%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 77.6

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

   11. Simplified 77.6%

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

   if 2.0000000000000001e-84 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 93.2

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

   5. Simplified 93.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 90.8

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

   8. Simplified 90.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 86.7

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

   10. Applied egg-rr 86.7%

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

   11. Taylor expanded in theta around 0

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

       1. cos-neg N/A

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

       2. +-inverses 86.7

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0} \cdot 0.5}{\cos delta} \]

   13. Simplified 86.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0} \cdot 0.5}{\cos delta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{0}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (0.5 - (0.5 * cos((phi1 + phi1)))))) + 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 = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (0.5d0 - (0.5d0 * cos((phi1 + phi1)))))) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 94.1

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

5. Simplified 94.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-atan2.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 94.1

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

7. Applied egg-rr 94.1%

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

Alternative 6: 89.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.3

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

5. Simplified 90.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 90.3

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

7. Applied egg-rr 90.3%

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

8. Final simplification 90.3%

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

Alternative 7: 86.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.3

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

5. Simplified 90.3%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 88.2

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

8. Simplified 88.2%

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

Alternative 8: 79.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, theta \cdot theta, 1\right) \cdot \sin delta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 6.2e-8)
   (+
    lambda1
    (atan2 (* (sin delta) (sin theta)) (fma -0.5 (* delta delta) 1.0)))
   (+
    lambda1
    (atan2
     (* theta (* (fma -0.16666666666666666 (* theta theta) 1.0) (sin delta)))
     (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 6.2e-8) {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), fma(-0.5, (delta * delta), 1.0));
	} else {
		tmp = lambda1 + atan2((theta * (fma(-0.16666666666666666, (theta * theta), 1.0) * sin(delta))), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 6.2e-8)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), fma(-0.5, Float64(delta * delta), 1.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * Float64(fma(-0.16666666666666666, Float64(theta * theta), 1.0) * sin(delta))), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 6.2e-8], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[(N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < 6.2e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.8

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

   5. Simplified 90.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.5

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

   8. Simplified 89.5%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 85.9

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

   11. Simplified 85.9%

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

   if 6.2e-8 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.0

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

   5. Simplified 89.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 84.5

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

   8. Simplified 84.5%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sin.f64 76.6

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, theta \cdot theta, 1\right) \cdot \color{blue}{\sin delta}\right)}{\cos delta} \]

   11. Simplified 76.6%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, theta \cdot theta, 1\right) \cdot \sin delta\right)}}{\cos delta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 8.8e-8)
   (+
    lambda1
    (atan2 (* (sin delta) (sin theta)) (fma -0.5 (* delta delta) 1.0)))
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 8.8e-8) {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), fma(-0.5, (delta * delta), 1.0));
	} else {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 8.8e-8)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), fma(-0.5, Float64(delta * delta), 1.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 8.8e-8], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < 8.7999999999999994e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.8

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

   5. Simplified 90.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.5

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

   8. Simplified 89.5%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 85.9

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

   11. Simplified 85.9%

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

   if 8.7999999999999994e-8 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.0

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

   5. Simplified 89.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 84.5

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

   8. Simplified 84.5%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 76.4

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

   11. Simplified 76.4%

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

Alternative 10: 80.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{if}\;theta \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 2.5 \cdot 10^{-57}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))))
   (if (<= theta -7.8e+18)
     t_1
     (if (<= theta 2.5e-57)
       (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((delta * sin(theta)), cos(delta));
	double tmp;
	if (theta <= -7.8e+18) {
		tmp = t_1;
	} else if (theta <= 2.5e-57) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((delta * sin(theta)), cos(delta))
    if (theta <= (-7.8d+18)) then
        tmp = t_1
    else if (theta <= 2.5d-57) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	double tmp;
	if (theta <= -7.8e+18) {
		tmp = t_1;
	} else if (theta <= 2.5e-57) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	tmp = 0
	if theta <= -7.8e+18:
		tmp = t_1
	elif theta <= 2.5e-57:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)))
	tmp = 0.0
	if (theta <= -7.8e+18)
		tmp = t_1;
	elseif (theta <= 2.5e-57)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((delta * sin(theta)), cos(delta));
	tmp = 0.0;
	if (theta <= -7.8e+18)
		tmp = t_1;
	elseif (theta <= 2.5e-57)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -7.8e+18], t$95$1, If[LessEqual[theta, 2.5e-57], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -7.8e18 or 2.5000000000000001e-57 < theta

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 86.5

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

   5. Simplified 86.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 84.8

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

   8. Simplified 84.8%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 76.8

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

   11. Simplified 76.8%

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

   if -7.8e18 < theta < 2.5000000000000001e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 91.8

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

   8. Simplified 91.8%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 91.7

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

   11. Simplified 91.7%

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

Alternative 11: 73.2% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2(0.0d0, cos(delta))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[0.0 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.3

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

5. Simplified 90.3%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 88.2

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

8. Simplified 88.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-+.f64 74.8

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

10. Applied egg-rr 74.8%

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

11. Taylor expanded in theta around 0

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

    1. cos-neg N/A

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

    2. +-inverses 74.7

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0} \cdot 0.5}{\cos delta} \]

13. Simplified 74.7%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0} \cdot 0.5}{\cos delta} \]

14. Final simplification 74.7%

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

Alternative 12: 70.2% accurate, 1341.0× 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 99.8%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in lambda1 around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\lambda_1}}} \]
6. Step-by-step derivation

   1. lower-/.f64 72.2

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\lambda_1}}} \]

7. Simplified 72.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\lambda_1}}} \]
8. Step-by-step derivation

   1. remove-double-div 72.3

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

9. Applied egg-rr 72.3%

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

Reproduce

herbie shell --seed 2024199 
(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))))))))))