Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.9%

Time: 15.6s

Alternatives: 11

Speedup: 1.3×

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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2 N/A

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

   8. sqr-cos-a N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(cos(delta), fma(0.5, cos(Float64(phi1 + phi1)), 0.5), Float64(Float64(-cos(phi1)) * Float64(Float64(cos(theta) * sin(phi1)) * sin(delta))))) + 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[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] + N[((-N[Cos[phi1], $MachinePrecision]) * N[(N[(N[Cos[theta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(Float64(Float64(cos(theta) * sin(phi1)) * Float64(-sin(delta))), cos(phi1), Float64(fma(cos(Float64(phi1 + phi1)), 0.5, 0.5) * cos(delta)))) + 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[(N[Cos[theta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Sin[delta], $MachinePrecision])), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 94.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(cos(delta), fma(0.5, cos(Float64(phi1 + phi1)), 0.5), Float64(Float64(Float64(-sin(delta)) * sin(phi1)) * cos(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[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[((-N[Sin[delta], $MachinePrecision]) * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in theta around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-sin.f64 N/A

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

    6. lower-sin.f64 94.8

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

11. Applied rewrites 94.8%

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

12. Final simplification 94.8%

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

Alternative 5: 92.7% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  (atan2
   (* (cos phi1) (* (sin delta) (sin theta)))
   (- (cos delta) (- 0.5 (* (cos (+ phi1 phi1)) 0.5))))
  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 - (cos((phi1 + phi1)) * 0.5)))) + 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 - (cos((phi1 + phi1)) * 0.5d0)))) + 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 - (Math.cos((phi1 + phi1)) * 0.5)))) + 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 - (math.cos((phi1 + phi1)) * 0.5)))) + 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(cos(Float64(phi1 + phi1)) * 0.5)))) + lambda1)
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (0.5 - (cos((phi1 + phi1)) * 0.5)))) + 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[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $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 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 92.4

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

7. Applied rewrites 92.4%

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

8. Final simplification 92.4%

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

Alternative 6: 89.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $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 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 89.4

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

5. Applied rewrites 89.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 89.4

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

7. Applied rewrites 89.4%

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

8. Final simplification 89.4%

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

Alternative 7: 87.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $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 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 89.4

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

5. Applied rewrites 89.4%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 87.4

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

8. Applied rewrites 87.4%

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

9. Final simplification 87.4%

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

Alternative 8: 79.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -2700000000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot theta, theta, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot theta\right) \cdot \sin delta}{1 - \phi_1 \cdot \phi_1} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -2700000000.0)
   (+
    (atan2
     (* (* (fma (* -0.16666666666666666 theta) theta 1.0) (sin delta)) theta)
     (cos delta))
    lambda1)
   (if (<= delta 2.4e+73)
     (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)
     (+
      (atan2 (* (* (cos phi1) theta) (sin delta)) (- 1.0 (* phi1 phi1)))
      lambda1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -2700000000.0) {
		tmp = atan2(((fma((-0.16666666666666666 * theta), theta, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1;
	} else if (delta <= 2.4e+73) {
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	} else {
		tmp = atan2(((cos(phi1) * theta) * sin(delta)), (1.0 - (phi1 * phi1))) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -2700000000.0)
		tmp = Float64(atan(Float64(Float64(fma(Float64(-0.16666666666666666 * theta), theta, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1);
	elseif (delta <= 2.4e+73)
		tmp = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1);
	else
		tmp = Float64(atan(Float64(Float64(cos(phi1) * theta) * sin(delta)), Float64(1.0 - Float64(phi1 * phi1))) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -2700000000.0], N[(N[ArcTan[N[(N[(N[(N[(-0.16666666666666666 * theta), $MachinePrecision] * theta + 1.0), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[delta, 2.4e+73], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * theta), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -2.7e9

   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 87.9

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

   5. Applied rewrites 87.9%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 71.5%

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

   if -2.7e9 < delta < 2.40000000000000002e73

   1. Initial program 99.9%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   2. 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 93.8

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

   5. Applied rewrites 93.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 93.4

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

   8. Applied rewrites 93.4%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 92.3%

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

   if 2.40000000000000002e73 < delta

   1. Initial program 99.4%

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

   3. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 56.8

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in phi1 around 0

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

   11. Applied rewrites 65.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2700000000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot theta, theta, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot theta\right) \cdot \sin delta}{1 - \phi_1 \cdot \phi_1} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -3100000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot theta\right) \cdot \sin delta}{1 - \phi_1 \cdot \phi_1} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -3100000000.0)
   (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
   (if (<= delta 2.4e+73)
     (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)
     (+
      (atan2 (* (* (cos phi1) theta) (sin delta)) (- 1.0 (* phi1 phi1)))
      lambda1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -3100000000.0) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} else if (delta <= 2.4e+73) {
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	} else {
		tmp = atan2(((cos(phi1) * theta) * sin(delta)), (1.0 - (phi1 * phi1))) + lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= (-3100000000.0d0)) then
        tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1
    else if (delta <= 2.4d+73) then
        tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1
    else
        tmp = atan2(((cos(phi1) * theta) * sin(delta)), (1.0d0 - (phi1 * phi1))) + lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -3100000000.0) {
		tmp = Math.atan2((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	} else if (delta <= 2.4e+73) {
		tmp = Math.atan2((delta * Math.sin(theta)), Math.cos(delta)) + lambda1;
	} else {
		tmp = Math.atan2(((Math.cos(phi1) * theta) * Math.sin(delta)), (1.0 - (phi1 * phi1))) + lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -3100000000.0:
		tmp = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	elif delta <= 2.4e+73:
		tmp = math.atan2((delta * math.sin(theta)), math.cos(delta)) + lambda1
	else:
		tmp = math.atan2(((math.cos(phi1) * theta) * math.sin(delta)), (1.0 - (phi1 * phi1))) + lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -3100000000.0)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	elseif (delta <= 2.4e+73)
		tmp = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1);
	else
		tmp = Float64(atan(Float64(Float64(cos(phi1) * theta) * sin(delta)), Float64(1.0 - Float64(phi1 * phi1))) + lambda1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -3100000000.0)
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	elseif (delta <= 2.4e+73)
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	else
		tmp = atan2(((cos(phi1) * theta) * sin(delta)), (1.0 - (phi1 * phi1))) + lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -3100000000.0], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[delta, 2.4e+73], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * theta), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -3.1e9

   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 87.9

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

   5. Applied rewrites 87.9%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 70.4%

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

   if -3.1e9 < delta < 2.40000000000000002e73

   1. Initial program 99.9%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   2. 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 93.8

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

   5. Applied rewrites 93.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 93.4

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

   8. Applied rewrites 93.4%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 92.3%

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

   if 2.40000000000000002e73 < delta

   1. Initial program 99.4%

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

   3. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 56.8

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in phi1 around 0

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

   11. Applied rewrites 65.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3100000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot theta\right) \cdot \sin delta}{1 - \phi_1 \cdot \phi_1} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -950000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)))
   (if (<= theta -950000.0)
     t_1
     (if (<= theta 1.4e+33)
       (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	double tmp;
	if (theta <= -950000.0) {
		tmp = t_1;
	} else if (theta <= 1.4e+33) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} 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 = atan2((delta * sin(theta)), cos(delta)) + lambda1
    if (theta <= (-950000.0d0)) then
        tmp = t_1
    else if (theta <= 1.4d+33) then
        tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1
    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 = Math.atan2((delta * Math.sin(theta)), Math.cos(delta)) + lambda1;
	double tmp;
	if (theta <= -950000.0) {
		tmp = t_1;
	} else if (theta <= 1.4e+33) {
		tmp = Math.atan2((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((delta * math.sin(theta)), math.cos(delta)) + lambda1
	tmp = 0
	if theta <= -950000.0:
		tmp = t_1
	elif theta <= 1.4e+33:
		tmp = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (theta <= -950000.0)
		tmp = t_1;
	elseif (theta <= 1.4e+33)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (theta <= -950000.0)
		tmp = t_1;
	elseif (theta <= 1.4e+33)
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -950000.0], t$95$1, If[LessEqual[theta, 1.4e+33], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -9.5e5 or 1.4e33 < theta

   1. Initial program 99.7%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   2. 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 86.2

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

   5. Applied rewrites 86.2%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 85.2

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.4%

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

   if -9.5e5 < theta < 1.4e33

   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 92.8

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

   5. Applied rewrites 92.8%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 89.8

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 89.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -950000:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $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 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 89.4

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

5. Applied rewrites 89.4%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 87.4

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

8. Applied rewrites 87.4%

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

9. Taylor expanded in theta around 0

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

11. Applied rewrites 72.3%

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

12. Final simplification 72.3%

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

Reproduce

herbie shell --seed 2024240 
(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))))))))))