Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 11.5s

Alternatives: 13

Speedup: N/A×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 94.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision]) + N[Cos[delta], $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--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 94.5

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

7. Applied rewrites 94.5%

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

8. Final simplification 94.5%

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

Alternative 3: 94.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 94.4

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

5. Applied rewrites 94.4%

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

6. Final simplification 94.4%

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

Alternative 4: 94.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 94.4

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

5. Applied rewrites 94.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 94.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 94.4

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

7. Applied rewrites 94.4%

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

8. Final simplification 94.4%

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

Alternative 5: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{\cos delta - \left(0.5 - \cos \left(\phi_1 + \phi_1\right) \cdot 0.5\right)} + \lambda_1\\ \mathbf{if}\;theta \leq -0.0185:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot theta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{fma}\left(\cos \phi_1, \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2
           (* (* (sin delta) (sin theta)) (cos phi1))
           (- (cos delta) (- 0.5 (* (cos (+ phi1 phi1)) 0.5))))
          lambda1)))
   (if (<= theta -0.0185)
     t_1
     (if (<= theta 8e-5)
       (+
        (atan2
         (* (* (sin delta) theta) (cos phi1))
         (-
          (cos delta)
          (*
           (fma (cos phi1) (sin delta) (* (sin phi1) (cos delta)))
           (sin phi1))))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((sin(delta) * sin(theta)) * cos(phi1)), (cos(delta) - (0.5 - (cos((phi1 + phi1)) * 0.5)))) + lambda1;
	double tmp;
	if (theta <= -0.0185) {
		tmp = t_1;
	} else if (theta <= 8e-5) {
		tmp = atan2(((sin(delta) * theta) * cos(phi1)), (cos(delta) - (fma(cos(phi1), sin(delta), (sin(phi1) * cos(delta))) * sin(phi1)))) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(sin(delta) * sin(theta)) * cos(phi1)), Float64(cos(delta) - Float64(0.5 - Float64(cos(Float64(phi1 + phi1)) * 0.5)))) + lambda1)
	tmp = 0.0
	if (theta <= -0.0185)
		tmp = t_1;
	elseif (theta <= 8e-5)
		tmp = Float64(atan(Float64(Float64(sin(delta) * theta) * cos(phi1)), Float64(cos(delta) - Float64(fma(cos(phi1), sin(delta), Float64(sin(phi1) * cos(delta))) * sin(phi1)))) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $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]}, If[LessEqual[theta, -0.0185], t$95$1, If[LessEqual[theta, 8e-5], N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -0.0184999999999999991 or 8.00000000000000065e-5 < 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 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 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 90.6%

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

   if -0.0184999999999999991 < theta < 8.00000000000000065e-5

   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 theta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in theta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -0.0185:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{\cos delta - \left(0.5 - \cos \left(\phi_1 + \phi_1\right) \cdot 0.5\right)} + \lambda_1\\ \mathbf{elif}\;theta \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot theta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{fma}\left(\cos \phi_1, \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{\cos delta - \left(0.5 - \cos \left(\phi_1 + \phi_1\right) \cdot 0.5\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return atan2(((sin(delta) * sin(theta)) * cos(phi1)), (cos(delta) - ((cos(((delta + phi1) - phi1)) - cos(((delta + phi1) + phi1))) / 2.0))) + 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(phi1)), (cos(delta) - ((cos(((delta + phi1) - phi1)) - cos(((delta + phi1) + phi1))) / 2.0d0))) + 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(phi1)), (Math.cos(delta) - ((Math.cos(((delta + phi1) - phi1)) - Math.cos(((delta + phi1) + phi1))) / 2.0))) + lambda1;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[(N[Cos[N[(N[(delta + phi1), $MachinePrecision] - phi1), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(N[(delta + phi1), $MachinePrecision] + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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 theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 94.4

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

5. Applied rewrites 94.4%

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

7. Applied rewrites 91.1%

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

8. Final simplification 91.1%

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

Alternative 7: 92.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  (atan2
   (* (* (sin delta) (sin theta)) (cos phi1))
   (- (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(((sin(delta) * sin(theta)) * cos(phi1)), (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(((sin(delta) * sin(theta)) * cos(phi1)), (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.sin(delta) * Math.sin(theta)) * Math.cos(phi1)), (Math.cos(delta) - (0.5 - (Math.cos((phi1 + phi1)) * 0.5)))) + lambda1;
}

Python

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $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 91.0

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

5. Applied rewrites 91.0%

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

7. Applied rewrites 91.1%

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

8. Final simplification 91.1%

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

Alternative 8: 88.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $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 89.2

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

5. Applied rewrites 89.2%

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

6. Final simplification 89.2%

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

Alternative 9: 88.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $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 89.2

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

5. Applied rewrites 89.2%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta}} \]
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 89.2

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

7. Applied rewrites 89.2%

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

8. Final simplification 89.2%

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

Alternative 10: 86.5% 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 89.2

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

5. Applied rewrites 89.2%

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

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 87.1

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

8. Applied rewrites 87.1%

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

9. Final simplification 87.1%

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

Alternative 11: 80.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -6.2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 1.85 \cdot 10^{-34}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2
           (*
            (* (fma (* -0.16666666666666666 delta) delta 1.0) (sin theta))
            delta)
           (cos delta))
          lambda1)))
   (if (<= theta -6.2e-12)
     t_1
     (if (<= theta 1.85e-34)
       (+ (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(((fma((-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1;
	double tmp;
	if (theta <= -6.2e-12) {
		tmp = t_1;
	} else if (theta <= 1.85e-34) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1)
	tmp = 0.0
	if (theta <= -6.2e-12)
		tmp = t_1;
	elseif (theta <= 1.85e-34)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -6.2e-12], t$95$1, If[LessEqual[theta, 1.85e-34], 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 < -6.2000000000000002e-12 or 1.84999999999999994e-34 < 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 85.8

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

   5. Applied rewrites 85.8%

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

   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 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 73.9%

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

   if -6.2000000000000002e-12 < theta < 1.84999999999999994e-34

   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 93.7

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

   5. Applied rewrites 93.7%

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

   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 90.6

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

   8. Applied rewrites 90.6%

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

   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 90.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -6.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 1.85 \cdot 10^{-34}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -5.2 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)))
   (if (<= delta -5.2e+71)
     t_1
     (if (<= delta 1.05e-26)
       (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -5.2e+71) {
		tmp = t_1;
	} else if (delta <= 1.05e-26) {
		tmp = atan2((delta * sin(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((sin(delta) * theta), cos(delta)) + lambda1
    if (delta <= (-5.2d+71)) then
        tmp = t_1
    else if (delta <= 1.05d-26) then
        tmp = atan2((delta * sin(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((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -5.2e+71) {
		tmp = t_1;
	} else if (delta <= 1.05e-26) {
		tmp = Math.atan2((delta * Math.sin(theta)), Math.cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -5.2e+71:
		tmp = t_1
	elif delta <= 1.05e-26:
		tmp = math.atan2((delta * math.sin(theta)), math.cos(delta)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -5.2e+71)
		tmp = t_1;
	elseif (delta <= 1.05e-26)
		tmp = Float64(atan(Float64(delta * sin(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((sin(delta) * theta), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -5.2e+71)
		tmp = t_1;
	elseif (delta <= 1.05e-26)
		tmp = atan2((delta * sin(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[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -5.2e+71], t$95$1, If[LessEqual[delta, 1.05e-26], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -5.19999999999999983e71 or 1.05000000000000004e-26 < delta

   1. Initial program 99.7%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   2. 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 83.3

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

   5. Applied rewrites 83.3%

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

   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. *-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 79.3

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

   8. Applied rewrites 79.3%

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

   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 65.8%

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

   if -5.19999999999999983e71 < delta < 1.05000000000000004e-26

   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 94.3

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

   5. Applied rewrites 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. *-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 93.9

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

   8. Applied rewrites 93.9%

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

   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 93.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -5.2 \cdot 10^{+71}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.0% 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 89.2

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

5. Applied rewrites 89.2%

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

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 87.1

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

8. Applied rewrites 87.1%

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

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 72.8%

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

12. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2024312 
(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))))))))))