Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.8%

Time: 15.6s

Alternatives: 13

Speedup: 1.2×

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(Float64(-(sin(phi1) ^ 2.0)), cos(delta), fma(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)), Float64(-sin(phi1)), 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[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]) * N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $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
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%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-lft-neg-out N/A

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

   4. lower-neg.f64 N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 99.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  (atan2
   (* (cos phi1) (* (sin delta) (sin theta)))
   (fma
    (- (* (cos (+ phi1 phi1)) 0.5) 0.5)
    (cos delta)
    (fma
     (* (* (cos phi1) (sin delta)) (cos theta))
     (- (sin phi1))
     (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((phi1 + phi1)) * 0.5) - 0.5), cos(delta), fma(((cos(phi1) * sin(delta)) * cos(theta)), -sin(phi1), 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(Float64(phi1 + phi1)) * 0.5) - 0.5), cos(delta), fma(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)), Float64(-sin(phi1)), 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[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $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
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%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-lft-neg-out N/A

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

   4. lower-neg.f64 N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 99.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

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

5. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), Float64(cos(delta) - Float64(fma(Float64(cos(theta) * sin(delta)), cos(phi1), Float64(cos(delta) * sin(phi1))) * sin(phi1)))) + 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[(N[Cos[delta], $MachinePrecision] - N[(N[(N[(N[Cos[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $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 delta around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 85.2

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

5. Applied rewrites 85.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 85.2

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

7. Applied rewrites 85.2%

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

8. Taylor expanded in theta around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 94.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 95.4

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

5. Applied rewrites 95.4%

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

6. Final simplification 95.4%

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

Alternative 6: 94.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), Float64(cos(delta) - Float64(fma(sin(phi1), cos(delta), Float64(cos(phi1) * sin(delta))) * sin(phi1)))) + 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[(N[Cos[delta], $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Sin[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 delta around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 85.2

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

5. Applied rewrites 85.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 85.2

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

7. Applied rewrites 85.2%

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

8. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 95.4

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

10. Applied rewrites 95.4%

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

11. Final simplification 95.4%

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

Alternative 7: 91.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $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 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 91.4

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

5. Applied rewrites 91.4%

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

6. Final simplification 91.4%

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

Alternative 8: 88.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $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 89.5

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

5. Applied rewrites 89.5%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta}} \]
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. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 89.5

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

7. Applied rewrites 89.5%

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

8. Final simplification 89.5%

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

Alternative 9: 85.8% 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 89.5

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

5. Applied rewrites 89.5%

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

6. Taylor expanded in phi1 around 0

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

   1. *-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 87.2

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

8. Applied rewrites 87.2%

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

9. Final simplification 87.2%

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

Alternative 10: 79.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(delta \cdot \sin theta\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right)}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -1.2 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 1.43 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2
           (*
            (* delta (sin theta))
            (fma (* -0.16666666666666666 delta) delta 1.0))
           (cos delta))
          lambda1)))
   (if (<= theta -1.2e-33)
     t_1
     (if (<= theta 1.43e-11)
       (+ (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)) * fma((-0.16666666666666666 * delta), delta, 1.0)), cos(delta)) + lambda1;
	double tmp;
	if (theta <= -1.2e-33) {
		tmp = t_1;
	} else if (theta <= 1.43e-11) {
		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(delta * sin(theta)) * fma(Float64(-0.16666666666666666 * delta), delta, 1.0)), cos(delta)) + lambda1)
	tmp = 0.0
	if (theta <= -1.2e-33)
		tmp = t_1;
	elseif (theta <= 1.43e-11)
		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[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -1.2e-33], t$95$1, If[LessEqual[theta, 1.43e-11], 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 < -1.2e-33 or 1.43000000000000001e-11 < 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 86.6

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

   5. Applied rewrites 86.6%

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

   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 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.7%

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

   if -1.2e-33 < theta < 1.43000000000000001e-11

   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 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 90.1

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

   8. Applied rewrites 90.1%

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

   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 90.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.2 \cdot 10^{-33}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(delta \cdot \sin theta\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right)}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 1.43 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(delta \cdot \sin theta\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right)}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := delta \cdot \sin theta\\ \mathbf{if}\;theta \leq -1.2 \cdot 10^{-33}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_1}{1} + \lambda_1\\ \mathbf{elif}\;theta \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* delta (sin theta))))
   (if (<= theta -1.2e-33)
     (+ (atan2 (* t_1 (cos phi1)) 1.0) lambda1)
     (if (<= theta 5e+79)
       (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
       (+ (atan2 t_1 (cos delta)) lambda1)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = delta * sin(theta);
	double tmp;
	if (theta <= -1.2e-33) {
		tmp = atan2((t_1 * cos(phi1)), 1.0) + lambda1;
	} else if (theta <= 5e+79) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} else {
		tmp = atan2(t_1, cos(delta)) + 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) :: t_1
    real(8) :: tmp
    t_1 = delta * sin(theta)
    if (theta <= (-1.2d-33)) then
        tmp = atan2((t_1 * cos(phi1)), 1.0d0) + lambda1
    else if (theta <= 5d+79) then
        tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1
    else
        tmp = atan2(t_1, cos(delta)) + lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = delta * Math.sin(theta);
	double tmp;
	if (theta <= -1.2e-33) {
		tmp = Math.atan2((t_1 * Math.cos(phi1)), 1.0) + lambda1;
	} else if (theta <= 5e+79) {
		tmp = Math.atan2((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	} else {
		tmp = Math.atan2(t_1, Math.cos(delta)) + lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = delta * math.sin(theta)
	tmp = 0
	if theta <= -1.2e-33:
		tmp = math.atan2((t_1 * math.cos(phi1)), 1.0) + lambda1
	elif theta <= 5e+79:
		tmp = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	else:
		tmp = math.atan2(t_1, math.cos(delta)) + lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(delta * sin(theta))
	tmp = 0.0
	if (theta <= -1.2e-33)
		tmp = Float64(atan(Float64(t_1 * cos(phi1)), 1.0) + lambda1);
	elseif (theta <= 5e+79)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = Float64(atan(t_1, cos(delta)) + lambda1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = delta * sin(theta);
	tmp = 0.0;
	if (theta <= -1.2e-33)
		tmp = atan2((t_1 * cos(phi1)), 1.0) + lambda1;
	elseif (theta <= 5e+79)
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	else
		tmp = atan2(t_1, cos(delta)) + lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.2e-33], N[(N[ArcTan[N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[theta, 5e+79], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if theta < -1.2e-33

   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. Step-by-step derivation

      1. lift-asin.f64 N/A

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

      2. asin-acos N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-acos.f64 99.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in delta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 78.1

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

   10. Applied rewrites 78.1%

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

   11. Taylor expanded in phi1 around 0

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

   13. Applied rewrites 72.5%

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

   if -1.2e-33 < theta < 5e79

   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 91.7

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

   5. Applied rewrites 91.7%

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

   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 89.3

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

   8. Applied rewrites 89.3%

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

   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 88.3%

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

   if 5e79 < 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 85.6

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

   8. Applied rewrites 85.6%

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

   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 81.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.2 \cdot 10^{-33}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(delta \cdot \sin theta\right) \cdot \cos \phi_1}{1} + \lambda_1\\ \mathbf{elif}\;theta \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)))
   (if (<= theta -5.2e-6)
     t_1
     (if (<= theta 5e+79)
       (+ (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 <= -5.2e-6) {
		tmp = t_1;
	} else if (theta <= 5e+79) {
		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 <= (-5.2d-6)) then
        tmp = t_1
    else if (theta <= 5d+79) 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 <= -5.2e-6) {
		tmp = t_1;
	} else if (theta <= 5e+79) {
		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 <= -5.2e-6:
		tmp = t_1
	elif theta <= 5e+79:
		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 <= -5.2e-6)
		tmp = t_1;
	elseif (theta <= 5e+79)
		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 <= -5.2e-6)
		tmp = t_1;
	elseif (theta <= 5e+79)
		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, -5.2e-6], t$95$1, If[LessEqual[theta, 5e+79], 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 < -5.20000000000000019e-6 or 5e79 < 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 86.8

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

   5. Applied rewrites 86.8%

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

   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 85.3

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

   8. Applied rewrites 85.3%

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

   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 74.9%

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

   if -5.20000000000000019e-6 < theta < 5e79

   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 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 88.6

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

   8. Applied rewrites 88.6%

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

   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 87.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.7% 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 89.5

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

5. Applied rewrites 89.5%

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

6. Taylor expanded in phi1 around 0

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

   1. *-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 87.2

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

8. Applied rewrites 87.2%

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

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 75.2%

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

12. Final simplification 75.2%

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

Reproduce

herbie shell --seed 2024242 
(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))))))))))