Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.9%

Time: 14.3s

Alternatives: 18

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \tan^{-1}_* \frac{t\_1 \cdot \sin theta}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot t\_1, \cos theta, {\cos \phi_1}^{2} \cdot \cos delta\right)} + \lambda_1 \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (+
    (atan2
     (* t_1 (sin theta))
     (fma
      (* (- (sin phi1)) t_1)
      (cos theta)
      (* (pow (cos phi1) 2.0) (cos delta))))
    lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * cos(phi1);
	return atan2((t_1 * sin(theta)), fma((-sin(phi1) * t_1), cos(theta), (pow(cos(phi1), 2.0) * cos(delta)))) + lambda1;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	return Float64(atan(Float64(t_1 * sin(theta)), fma(Float64(Float64(-sin(phi1)) * t_1), cos(theta), Float64((cos(phi1) ^ 2.0) * cos(delta)))) + lambda1)
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(N[ArcTan[N[(t$95$1 * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Sin[phi1], $MachinePrecision]) * t$95$1), $MachinePrecision] * N[Cos[theta], $MachinePrecision] + N[(N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $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%

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

5. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. 1-sub-sin N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.9

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

9. Applied rewrites 99.9%

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

11. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.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%

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

5. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. 1-sub-sin N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

5. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. 1-sub-sin N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.9

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

9. Applied rewrites 99.9%

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

11. Applied rewrites 99.9%

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

Alternative 4: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), fma(Float64(cos(theta) * Float64(sin(delta) * cos(phi1))), Float64(-sin(phi1)), Float64(fma(cos(Float64(2.0 * phi1)), 0.5, 0.5) * cos(delta)))))
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[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision]) + N[(N[(N[Cos[N[(2.0 * phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

5. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. 1-sub-sin N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

Alternative 5: 95.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

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

5. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. 1-sub-sin N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in theta around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-sin.f64 94.1

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

10. Applied rewrites 94.1%

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

Alternative 6: 95.0% accurate, 1.3× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.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%

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

5. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. 1-sub-sin N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in theta around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-sin.f64 N/A

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

    6. lower-sin.f64 94.0

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

12. Applied rewrites 94.0%

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

Alternative 7: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. Add Preprocessing
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. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 94.0

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

7. Applied rewrites 94.0%

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

Alternative 8: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-sin.f64 89.3

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

5. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.3

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 89.3

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

7. Applied rewrites 89.3%

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

8. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 94.0

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

10. Applied rewrites 94.0%

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

Alternative 9: 92.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. Add Preprocessing
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. Taylor expanded in delta around 0

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

   1. lower-sin.f64 92.6

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

7. Applied rewrites 92.6%

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

Alternative 10: 92.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-sin.f64 89.3

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

5. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 89.3

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 89.3

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

7. Applied rewrites 89.3%

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

8. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 92.6

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

10. Applied rewrites 92.6%

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

Alternative 11: 91.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -1450000 \lor \neg \left(delta \leq 3.5 \cdot 10^{-25}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot delta\right) \cdot \sin theta}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -1450000.0) (not (<= delta 3.5e-25)))
   (+ lambda1 (atan2 (* (* (cos phi1) (sin delta)) (sin theta)) (cos delta)))
   (+
    lambda1
    (atan2 (* (* (cos phi1) delta) (sin theta)) (* (cos phi1) (cos phi1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -1450000.0) || !(delta <= 3.5e-25)) {
		tmp = lambda1 + atan2(((cos(phi1) * sin(delta)) * sin(theta)), cos(delta));
	} else {
		tmp = lambda1 + atan2(((cos(phi1) * delta) * sin(theta)), (cos(phi1) * cos(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-1450000.0d0)) .or. (.not. (delta <= 3.5d-25))) then
        tmp = lambda1 + atan2(((cos(phi1) * sin(delta)) * sin(theta)), cos(delta))
    else
        tmp = lambda1 + atan2(((cos(phi1) * delta) * sin(theta)), (cos(phi1) * cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -1450000.0) || !(delta <= 3.5e-25)) {
		tmp = lambda1 + Math.atan2(((Math.cos(phi1) * Math.sin(delta)) * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(((Math.cos(phi1) * delta) * Math.sin(theta)), (Math.cos(phi1) * Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -1450000.0) or not (delta <= 3.5e-25):
		tmp = lambda1 + math.atan2(((math.cos(phi1) * math.sin(delta)) * math.sin(theta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(((math.cos(phi1) * delta) * math.sin(theta)), (math.cos(phi1) * math.cos(phi1)))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -1450000.0) || !(delta <= 3.5e-25))
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * sin(delta)) * sin(theta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * delta) * sin(theta)), Float64(cos(phi1) * cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -1450000.0) || ~((delta <= 3.5e-25)))
		tmp = lambda1 + atan2(((cos(phi1) * sin(delta)) * sin(theta)), cos(delta));
	else
		tmp = lambda1 + atan2(((cos(phi1) * delta) * sin(theta)), (cos(phi1) * cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -1450000.0], N[Not[LessEqual[delta, 3.5e-25]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * delta), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.45e6 or 3.5000000000000002e-25 < delta

   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 85.8

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

   5. Applied rewrites 85.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta}} \]
   6. 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 85.8

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

      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 85.8

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

   7. Applied rewrites 85.8%

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

   if -1.45e6 < delta < 3.5000000000000002e-25

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in delta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1450000 \lor \neg \left(delta \leq 3.5 \cdot 10^{-25}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot delta\right) \cdot \sin theta}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

   1. lift-*.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 88.7

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

   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 88.7

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

7. Applied rewrites 88.7%

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

Alternative 13: 86.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

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 85.7

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

8. Applied rewrites 85.7%

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

Alternative 14: 80.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -17 \lor \neg \left(delta \leq 7.1 \cdot 10^{-16}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \left(\sin delta \cdot theta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \left(\sin theta \cdot delta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -17.0) (not (<= delta 7.1e-16)))
   (+
    lambda1
    (atan2
     (* (fma (* theta theta) -0.16666666666666666 1.0) (* (sin delta) theta))
     (cos delta)))
   (+
    lambda1
    (atan2
     (* (fma (* -0.16666666666666666 delta) delta 1.0) (* (sin theta) delta))
     (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -17.0) || !(delta <= 7.1e-16)) {
		tmp = lambda1 + atan2((fma((theta * theta), -0.16666666666666666, 1.0) * (sin(delta) * theta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((fma((-0.16666666666666666 * delta), delta, 1.0) * (sin(theta) * delta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -17.0) || !(delta <= 7.1e-16))
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(theta * theta), -0.16666666666666666, 1.0) * Float64(sin(delta) * theta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * Float64(sin(theta) * delta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -17.0], N[Not[LessEqual[delta, 7.1e-16]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[(N[(theta * theta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -17 or 7.1e-16 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 85.2

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

   5. Applied rewrites 85.2%

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

   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 80.1

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 72.2%

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

   if -17 < delta < 7.1e-16

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 91.4

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

   8. Applied rewrites 91.4%

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

   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 91.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -17 \lor \neg \left(delta \leq 7.1 \cdot 10^{-16}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \left(\sin delta \cdot theta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \left(\sin theta \cdot delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot delta\\ \mathbf{if}\;theta \leq -1.02 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot t\_1}{\cos delta}\\ \mathbf{elif}\;theta \leq 3 \cdot 10^{-20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) delta)))
   (if (<= theta -1.02e+20)
     (+
      lambda1
      (atan2
       (* (fma (* -0.16666666666666666 delta) delta 1.0) t_1)
       (cos delta)))
     (if (<= theta 3e-20)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       (+ lambda1 (atan2 t_1 (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * delta;
	double tmp;
	if (theta <= -1.02e+20) {
		tmp = lambda1 + atan2((fma((-0.16666666666666666 * delta), delta, 1.0) * t_1), cos(delta));
	} else if (theta <= 3e-20) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * delta)
	tmp = 0.0
	if (theta <= -1.02e+20)
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * t_1), cos(delta)));
	elseif (theta <= 3e-20)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]}, If[LessEqual[theta, -1.02e+20], N[(lambda1 + N[ArcTan[N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[theta, 3e-20], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if theta < -1.02e20

   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 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 84.6

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

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.2%

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

   if -1.02e20 < theta < 3.00000000000000029e-20

   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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 90.5

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

   8. Applied rewrites 90.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 89.9%

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

   if 3.00000000000000029e-20 < theta

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 78.7

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

   8. Applied rewrites 78.7%

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

   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 73.5%

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

Alternative 16: 80.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -4.8 \cdot 10^{+19} \lor \neg \left(theta \leq 3 \cdot 10^{-20}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -4.8e+19) (not (<= theta 3e-20)))
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -4.8e+19) || !(theta <= 3e-20)) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((theta <= (-4.8d+19)) .or. (.not. (theta <= 3d-20))) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -4.8e+19) || !(theta <= 3e-20)) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (theta <= -4.8e+19) or not (theta <= 3e-20):
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -4.8e+19) || !(theta <= 3e-20))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((theta <= -4.8e+19) || ~((theta <= 3e-20)))
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -4.8e+19], N[Not[LessEqual[theta, 3e-20]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -4.8e19 or 3.00000000000000029e-20 < 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 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 81.3

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

   8. Applied rewrites 81.3%

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

   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 73.9%

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

   if -4.8e19 < theta < 3.00000000000000029e-20

   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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 90.5

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

   8. Applied rewrites 90.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 89.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -4.8 \cdot 10^{+19} \lor \neg \left(theta \leq 3 \cdot 10^{-20}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

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 85.7

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

8. Applied rewrites 85.7%

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

9. Taylor expanded in theta around 0

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

11. Applied rewrites 72.3%

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

Alternative 18: 67.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

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 85.7

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

8. Applied rewrites 85.7%

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

9. Taylor expanded in theta around 0

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

11. Applied rewrites 72.3%

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

12. Taylor expanded in delta around 0

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

14. Applied rewrites 64.8%

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

Reproduce

herbie shell --seed 2024314 
(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))))))))))