Midpoint on a great circle

Percentage Accurate: 98.6% → 99.6%

Time: 22.7s

Alternatives: 18

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (fma (sin (- lambda2)) (cos lambda1) (* (sin lambda1) (cos lambda2))))
   (fma
    (cos phi2)
    (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))
    (cos phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * fma(sin(-lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2)))), fma(cos(phi2), fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))), cos(phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * fma(sin(Float64(-lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2)))), fma(cos(phi2), fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))), cos(phi1))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. sin-sum N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. cos-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\cos \lambda_2} \cdot \sin \lambda_1\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. *-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. lower-neg.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   12. lower-cos.f64 98.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

4. Applied rewrites 98.6%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
5. Step-by-step derivation

   1. cos-diff N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]

   2. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

   3. lift-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

   4. lift-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

   5. *-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

   7. lift-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)} \]

   8. lift-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)} \]

   9. lower-*.f64 99.6

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)} \]

6. Applied rewrites 99.6%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

7. Taylor expanded in phi1 around inf

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \phi_1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1\right)}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}, \cos \phi_1\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\color{blue}{\cos \lambda_1}, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right), \cos \phi_1\right)} \]

   8. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right), \cos \phi_1\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right), \cos \phi_1\right)} \]

   10. lower-cos.f64 99.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \color{blue}{\cos \phi_1}\right)} \]

9. Applied rewrites 99.6%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1\right)}} \]
10. Add Preprocessing

Alternative 2: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t\_1\\ t_3 := \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \cos \phi_1\right)}\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos \phi_1 + \cos \phi_2 \cdot t\_0}\\ t_5 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), t\_0, \cos \phi_1\right)\\ \mathbf{if}\;t\_4 \leq -4000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \left(0.5 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right)}{t\_5}\\ \mathbf{elif}\;t\_4 \leq -0.02:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \lambda_2 \cdot \lambda_2, 1\right), \cos \phi_2, \cos \phi_1\right)}\\ \mathbf{elif}\;t\_4 \leq 3:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_5}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* (cos phi2) t_1))
        (t_3 (atan2 t_2 (fma (cos phi2) (cos lambda2) (cos phi1))))
        (t_4 (+ lambda1 (atan2 t_2 (+ (cos phi1) (* (cos phi2) t_0)))))
        (t_5 (fma (fma -0.5 (* phi2 phi2) 1.0) t_0 (cos phi1))))
   (if (<= t_4 -4000000.0)
     (+
      lambda1
      (atan2 (* (* (cos phi2) (sin lambda2)) (* 0.5 (* lambda1 lambda1))) t_5))
     (if (<= t_4 -0.02)
       t_3
       (if (<= t_4 0.002)
         (+
          lambda1
          (atan2
           t_2
           (fma (fma -0.5 (* lambda2 lambda2) 1.0) (cos phi2) (cos phi1))))
         (if (<= t_4 3.0) t_3 (+ lambda1 (atan2 t_1 t_5))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = cos(phi2) * t_1;
	double t_3 = atan2(t_2, fma(cos(phi2), cos(lambda2), cos(phi1)));
	double t_4 = lambda1 + atan2(t_2, (cos(phi1) + (cos(phi2) * t_0)));
	double t_5 = fma(fma(-0.5, (phi2 * phi2), 1.0), t_0, cos(phi1));
	double tmp;
	if (t_4 <= -4000000.0) {
		tmp = lambda1 + atan2(((cos(phi2) * sin(lambda2)) * (0.5 * (lambda1 * lambda1))), t_5);
	} else if (t_4 <= -0.02) {
		tmp = t_3;
	} else if (t_4 <= 0.002) {
		tmp = lambda1 + atan2(t_2, fma(fma(-0.5, (lambda2 * lambda2), 1.0), cos(phi2), cos(phi1)));
	} else if (t_4 <= 3.0) {
		tmp = t_3;
	} else {
		tmp = lambda1 + atan2(t_1, t_5);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * t_1)
	t_3 = atan(t_2, fma(cos(phi2), cos(lambda2), cos(phi1)))
	t_4 = Float64(lambda1 + atan(t_2, Float64(cos(phi1) + Float64(cos(phi2) * t_0))))
	t_5 = fma(fma(-0.5, Float64(phi2 * phi2), 1.0), t_0, cos(phi1))
	tmp = 0.0
	if (t_4 <= -4000000.0)
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi2) * sin(lambda2)) * Float64(0.5 * Float64(lambda1 * lambda1))), t_5));
	elseif (t_4 <= -0.02)
		tmp = t_3;
	elseif (t_4 <= 0.002)
		tmp = Float64(lambda1 + atan(t_2, fma(fma(-0.5, Float64(lambda2 * lambda2), 1.0), cos(phi2), cos(phi1))));
	elseif (t_4 <= 3.0)
		tmp = t_3;
	else
		tmp = Float64(lambda1 + atan(t_1, t_5));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$2 / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4000000.0], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(lambda1 * lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.02], t$95$3, If[LessEqual[t$95$4, 0.002], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[(-0.5 * N[(lambda2 * lambda2), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 3.0], t$95$3, N[(lambda1 + N[ArcTan[t$95$1 / t$95$5], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -4e6

   1. Initial program 99.1%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)} + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right)} + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \left(\lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \lambda_1} + \left(\lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. associate-*l* N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \lambda_1\right)} + \left(\lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} + \left(\lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right) + \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \color{blue}{\left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, -\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.5, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\color{blue}{\cos \phi_1 + \left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1}} \]

      2. associate-*r* N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) + \cos \phi_1} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      4. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}} \]

      6. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2} + 1}, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)}, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      8. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, \cos \phi_1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, \mathsf{neg}\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{2}, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}, \cos \phi_1\right)} \]

      12. lower-cos.f64 99.2

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, -\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.5, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \color{blue}{\cos \phi_1}\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \lambda_1 \cdot \cos \lambda_2, -\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.5, 1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}} \]

   9. Taylor expanded in lambda1 around inf

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\frac{1}{2} \cdot \left({\lambda_1}^{2} \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left({\lambda_1}^{2} \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right) \cdot \frac{1}{2}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot {\lambda_1}^{2}\right)} \cdot \frac{1}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       3. associate-*l* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \left({\lambda_1}^{2} \cdot \frac{1}{2}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       4. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_2 \cdot \sin \lambda_2\right)} \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       7. lower-cos.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\color{blue}{\cos \phi_2} \cdot \sin \lambda_2\right) \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       8. lower-sin.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \color{blue}{\sin \lambda_2}\right) \cdot \left(\frac{1}{2} \cdot {\lambda_1}^{2}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {\lambda_1}^{2}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       10. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

       11. lower-*.f64 99.2

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \left(0.5 \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

   11. Applied rewrites 99.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \left(0.5 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

   if -4e6 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < -0.0200000000000000004 or 2e-3 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 3

   1. Initial program 97.0%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 97.0

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 97.0%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2}} \]
   7. Step-by-step derivation

      1. lower-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2} \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2} \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2} \]

      5. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2} \]

      6. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_2 \cdot \cos \phi_2 + \cos \phi_1}} \]

      7. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \cos \phi_2 + \cos \phi_1} \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \phi_1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_1\right)}} \]

      10. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_1\right)} \]

      11. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \lambda_2}, \cos \phi_1\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \lambda_2}, \cos \phi_1\right)} \]

      13. lower-cos.f64 96.8

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \color{blue}{\cos \phi_1}\right)} \]

   8. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \cos \phi_1\right)}} \]

   if -0.0200000000000000004 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))) < 2e-3

   1. Initial program 99.0%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 97.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 97.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \left(\cos \phi_2 + \frac{-1}{2} \cdot \left({\lambda_2}^{2} \cdot \cos \phi_2\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(\cos \phi_2 + \frac{-1}{2} \cdot \left({\lambda_2}^{2} \cdot \cos \phi_2\right)\right) + \cos \phi_1}} \]

      2. associate-*r* N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(\cos \phi_2 + \color{blue}{\left(\frac{-1}{2} \cdot {\lambda_2}^{2}\right) \cdot \cos \phi_2}\right) + \cos \phi_1} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(\frac{-1}{2} \cdot {\lambda_2}^{2} + 1\right) \cdot \cos \phi_2} + \cos \phi_1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot {\lambda_2}^{2} + 1, \cos \phi_2, \cos \phi_1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\lambda_2}^{2}, 1\right)}, \cos \phi_2, \cos \phi_1\right)} \]

      6. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\lambda_2 \cdot \lambda_2}, 1\right), \cos \phi_2, \cos \phi_1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\lambda_2 \cdot \lambda_2}, 1\right), \cos \phi_2, \cos \phi_1\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \lambda_2 \cdot \lambda_2, 1\right), \color{blue}{\cos \phi_2}, \cos \phi_1\right)} \]

      9. lower-cos.f64 97.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \lambda_2 \cdot \lambda_2, 1\right), \cos \phi_2, \color{blue}{\cos \phi_1}\right)} \]

   8. Applied rewrites 97.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \lambda_2 \cdot \lambda_2, 1\right), \cos \phi_2, \cos \phi_1\right)}} \]

   if 3 < (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))))))

   1. Initial program 98.9%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower--.f64 98.9

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1}} \]

      2. associate-*r* N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) + \cos \phi_1} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      4. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}} \]

      6. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2} + 1}, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)}, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      8. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, \cos \phi_1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \phi_2 \cdot \phi_2, 1\right), \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}, \cos \phi_1\right)} \]

      12. lower-cos.f64 98.9

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \color{blue}{\cos \phi_1}\right)} \]

   8. Applied rewrites 98.9%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (fma (sin (- lambda2)) (cos lambda1) (* (sin lambda1) (cos lambda2))))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * fma(sin(-lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2)))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * fma(sin(Float64(-lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2)))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. sin-sum N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. cos-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\cos \lambda_2} \cdot \sin \lambda_1\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. *-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. lower-neg.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}, \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \cos \lambda_2}\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \cos \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   12. lower-cos.f64 98.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

4. Applied rewrites 98.6%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.99999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \phi_2, t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.99999)
     (+ lambda1 (atan2 t_1 (fma t_0 (fma -0.5 (* phi2 phi2) 1.0) (cos phi1))))
     (+ lambda1 (atan2 t_1 (fma (cos phi2) t_0 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.99999) {
		tmp = lambda1 + atan2(t_1, fma(t_0, fma(-0.5, (phi2 * phi2), 1.0), cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_1, fma(cos(phi2), t_0, 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.99999)
		tmp = Float64(lambda1 + atan(t_1, fma(t_0, fma(-0.5, Float64(phi2 * phi2), 1.0), cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(cos(phi2), t_0, 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.99999], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 * N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.999990000000000046

   1. Initial program 98.5%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1}} \]

      2. associate-*r* N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) + \cos \phi_1} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      4. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)} + \cos \phi_1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \frac{-1}{2} \cdot {\phi_2}^{2} + 1, \cos \phi_1\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, \frac{-1}{2} \cdot {\phi_2}^{2} + 1, \cos \phi_1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}, \frac{-1}{2} \cdot {\phi_2}^{2} + 1, \cos \phi_1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {\phi_2}^{2}, 1\right)}, \cos \phi_1\right)} \]

      9. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right), \cos \phi_1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\phi_2 \cdot \phi_2}, 1\right), \cos \phi_1\right)} \]

      11. lower-cos.f64 82.1

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \color{blue}{\cos \phi_1}\right)} \]

   5. Applied rewrites 82.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \cos \phi_1\right)}} \]

   if 0.999990000000000046 < (cos.f64 phi1)

   1. Initial program 98.4%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), 1\right)}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \left(\lambda_1 - \lambda_2\right), 1\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, 1\right)} \]

      5. lower--.f64 98.4

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}, 1\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.99999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{\mathsf{fma}\left(\cos \phi_2, t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (sin (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.99999)
     (+ lambda1 (atan2 t_1 (+ (cos phi1) (* (cos phi2) t_0))))
     (+ lambda1 (atan2 (* (cos phi2) t_1) (fma (cos phi2) t_0 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.99999) {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_1), fma(cos(phi2), t_0, 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.99999)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * t_0))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), fma(cos(phi2), t_0, 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.99999], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.999990000000000046

   1. Initial program 98.5%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower--.f64 81.3

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 81.3%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if 0.999990000000000046 < (cos.f64 phi1)

   1. Initial program 98.4%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), 1\right)}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \left(\lambda_1 - \lambda_2\right), 1\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, 1\right)} \]

      5. lower--.f64 98.4

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}, 1\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.99999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.99999)
     (+
      lambda1
      (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
     (+
      lambda1
      (atan2 (* (cos phi2) t_0) (fma (cos phi2) (cos lambda2) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.99999) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), fma(cos(phi2), cos(lambda2), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.99999)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), fma(cos(phi2), cos(lambda2), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.99999], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.999990000000000046

   1. Initial program 98.5%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower--.f64 81.3

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 81.3%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   if 0.999990000000000046 < (cos.f64 phi1)

   1. Initial program 98.4%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 98.1

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \lambda_2 \cdot \cos \phi_2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_2 \cdot \cos \phi_2 + 1}} \]

      2. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \cos \phi_2 + 1} \]

      3. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), 1\right)}} \]

      5. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), 1\right)} \]

      6. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \lambda_2}, 1\right)} \]

      7. lower-cos.f64 98.1

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \lambda_2}, 1\right)} \]

   8. Applied rewrites 98.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9999998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.9999998)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.9999998) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.9999998d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.9999998) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.9999998:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos((lambda1 - lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi2) <= 0.9999998)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.9999998)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.9999998], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.999999799999999994

   1. Initial program 97.8%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 97.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 97.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \phi_2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 + \cos \phi_1}} \]

      2. lower-+.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 + \cos \phi_1}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2} + \cos \phi_1} \]

      4. lower-cos.f64 78.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \color{blue}{\cos \phi_1}} \]

   8. Applied rewrites 78.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 + \cos \phi_1}} \]

   if 0.999999799999999994 < (cos.f64 phi2)

   1. Initial program 99.1%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

      2. lower-+.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      4. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      5. lower-cos.f64 98.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

   5. Applied rewrites 98.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.9999998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 9: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9999998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.9999998)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.9999998) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.9999998d0) then
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.9999998) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.9999998:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos((lambda1 - lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.9999998)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.9999998)
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.9999998], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.999999799999999994

   1. Initial program 97.8%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

      2. lower-cos.f64 97.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   5. Applied rewrites 97.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \phi_2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 + \cos \phi_1}} \]

      2. lower-+.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 + \cos \phi_1}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2} + \cos \phi_1} \]

      4. lower-cos.f64 78.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \color{blue}{\cos \phi_1}} \]

   8. Applied rewrites 78.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 + \cos \phi_1}} \]

   if 0.999999799999999994 < (cos.f64 phi2)

   1. Initial program 99.1%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower--.f64 98.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

      2. lower-+.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      4. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      5. lower-cos.f64 98.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

   8. Applied rewrites 98.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.9999998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \cos \phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (fma (cos phi2) (cos lambda2) (cos phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), fma(cos(phi2), cos(lambda2), cos(phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(cos(phi2), cos(lambda2), cos(phi1))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in lambda1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \cos \phi_1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_1\right)}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_1\right)} \]

   4. cos-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \lambda_2}, \cos \phi_1\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \lambda_2}, \cos \phi_1\right)} \]

   6. lower-cos.f64 98.0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \color{blue}{\cos \phi_1}\right)} \]

5. Applied rewrites 98.0%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2, \cos \phi_1\right)}} \]
6. Add Preprocessing

Alternative 11: 71.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.992715:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), \cos \left(\lambda_1 - \lambda_2\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.992715)
     (+ lambda1 (atan2 t_0 (+ (cos lambda1) (cos phi1))))
     (+
      lambda1
      (atan2
       t_0
       (fma
        (* phi1 phi1)
        (fma
         phi1
         (*
          phi1
          (fma phi1 (* phi1 -0.001388888888888889) 0.041666666666666664))
         -0.5)
        (+ (cos (- lambda1 lambda2)) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.992715) {
		tmp = lambda1 + atan2(t_0, (cos(lambda1) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, fma((phi1 * phi1), fma(phi1, (phi1 * fma(phi1, (phi1 * -0.001388888888888889), 0.041666666666666664)), -0.5), (cos((lambda1 - lambda2)) + 1.0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.992715)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(lambda1) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, fma(Float64(phi1 * phi1), fma(phi1, Float64(phi1 * fma(phi1, Float64(phi1 * -0.001388888888888889), 0.041666666666666664)), -0.5), Float64(cos(Float64(lambda1 - lambda2)) + 1.0))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.992715], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[lambda1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[(phi1 * phi1), $MachinePrecision] * N[(phi1 * N[(phi1 * N[(phi1 * N[(phi1 * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99271500000000001

   1. Initial program 98.5%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower--.f64 80.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 80.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

      2. lower-+.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

      3. lower-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      4. lower--.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

      5. lower-cos.f64 79.3

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

   8. Applied rewrites 79.3%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_1} + \cos \phi_1} \]
   10. Step-by-step derivation

       1. lower-cos.f64 70.1

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_1} + \cos \phi_1} \]

   11. Applied rewrites 70.1%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_1} + \cos \phi_1} \]

   if 0.99271500000000001 < (cos.f64 phi1)

   1. Initial program 98.5%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower--.f64 79.2

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Applied rewrites 79.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\left(1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{{\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) + \left(1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left({\phi_1}^{2}, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{{\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\left(\phi_1 \cdot \phi_1\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right)\right) + \color{blue}{\frac{-1}{2}}, 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\mathsf{fma}\left(\phi_1, \phi_1 \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right), \frac{-1}{2}\right)}, 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \color{blue}{\phi_1 \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right)}, \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \color{blue}{\left(\frac{-1}{720} \cdot {\phi_1}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \left(\color{blue}{{\phi_1}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      14. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \left(\color{blue}{\left(\phi_1 \cdot \phi_1\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      15. associate-*l* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \left(\color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_1, \phi_1 \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \color{blue}{\phi_1 \cdot \frac{-1}{720}}, \frac{1}{24}\right), \frac{-1}{2}\right), 1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   8. Applied rewrites 79.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), \mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), 1\right)\right)}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), \color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), \color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

       2. lower-cos.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

       3. lower--.f64 79.2

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)} \]

   11. Applied rewrites 79.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), \color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.992715:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot \mathsf{fma}\left(\phi_1, \phi_1 \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), \cos \left(\lambda_1 - \lambda_2\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos lambda2) (cos phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(lambda2) + Math.cos(phi1)));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(lambda2) + math.cos(phi1)))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(lambda2) + cos(phi1))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in lambda1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
4. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

   2. lower-cos.f64 98.0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

5. Applied rewrites 98.0%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\cos \lambda_2}} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_2 + \cos \phi_1}} \]
7. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \phi_1} \]

   2. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]

   3. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]

   4. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1} + \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \]

   5. cos-neg N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \color{blue}{\cos \lambda_2}} \]

   6. lower-cos.f64 80.1

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \color{blue}{\cos \lambda_2}} \]

8. Applied rewrites 80.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \lambda_2}} \]

9. Final simplification 80.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1} \]
10. Add Preprocessing

Alternative 13: 76.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2 (sin (- lambda1 lambda2)) (+ (cos phi1) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda1 - lambda2))));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda1 - lambda2))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(phi1) + Math.cos((lambda1 - lambda2))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(phi1) + math.cos((lambda1 - lambda2))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + cos(Float64(lambda1 - lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + cos((lambda1 - lambda2))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. lower--.f64 79.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 79.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   2. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   4. lower--.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   5. lower-cos.f64 79.3

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

8. Applied rewrites 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

9. Final simplification 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
10. Add Preprocessing

Alternative 14: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 (sin (- lambda1 lambda2)) (+ (cos lambda2) (cos phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda2) + cos(phi1)));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda2) + cos(phi1)))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(lambda2) + Math.cos(phi1)));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(lambda2) + math.cos(phi1)))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(lambda2) + cos(phi1))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda2) + cos(phi1)));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. lower--.f64 79.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 79.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   2. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   4. lower--.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   5. lower-cos.f64 79.3

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

8. Applied rewrites 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

9. Taylor expanded in lambda1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \cos \phi_1} \]
10. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_2} + \cos \phi_1} \]

    2. lower-cos.f64 78.8

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_2} + \cos \phi_1} \]

11. Applied rewrites 78.8%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \lambda_2} + \cos \phi_1} \]
12. Add Preprocessing

Alternative 15: 66.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2 (sin (- lambda1 lambda2)) (+ (cos (- lambda1 lambda2)) 1.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) + 1.0));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) + 1.0d0))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos((lambda1 - lambda2)) + 1.0));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos((lambda1 - lambda2)) + 1.0))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(Float64(lambda1 - lambda2)) + 1.0)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) + 1.0));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. lower--.f64 79.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 79.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   2. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   4. lower--.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   5. lower-cos.f64 79.3

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

8. Applied rewrites 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

9. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

    2. lower-cos.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

    3. lower--.f64 70.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

11. Applied rewrites 70.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

12. Final simplification 70.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1} \]
13. Add Preprocessing

Alternative 16: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + 1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 (sin (- lambda1 lambda2)) (+ (cos lambda2) 1.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda2) + 1.0));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda2) + 1.0d0))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(lambda2) + 1.0));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(lambda2) + 1.0))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(lambda2) + 1.0)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda2) + 1.0));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. lower--.f64 79.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 79.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   2. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   4. lower--.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   5. lower-cos.f64 79.3

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

8. Applied rewrites 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

9. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

    2. lower-cos.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

    3. lower--.f64 70.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

11. Applied rewrites 70.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

12. Taylor expanded in lambda1 around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
13. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \lambda_2}} \]

    2. lower-cos.f64 70.4

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \lambda_2}} \]

14. Applied rewrites 70.4%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \lambda_2}} \]

15. Final simplification 70.4%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + 1} \]
16. Add Preprocessing

Alternative 17: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 + 1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 (sin (- lambda1 lambda2)) (+ (cos lambda1) 1.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda1) + 1.0));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda1) + 1.0d0))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(lambda1) + 1.0));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(lambda1) + 1.0))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(lambda1) + 1.0)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda1) + 1.0));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. lower--.f64 79.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 79.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   2. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   4. lower--.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   5. lower-cos.f64 79.3

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

8. Applied rewrites 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

9. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

    2. lower-cos.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

    3. lower--.f64 70.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

11. Applied rewrites 70.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

12. Taylor expanded in lambda2 around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \lambda_1}} \]
13. Step-by-step derivation

    1. lower-cos.f64 64.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \lambda_1}} \]

14. Applied rewrites 64.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \lambda_1}} \]

15. Final simplification 64.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 + 1} \]
16. Add Preprocessing

Alternative 18: 55.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \left(\lambda_1 - \lambda_2\right) + 1} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 (sin lambda1) (+ (cos (- lambda1 lambda2)) 1.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin(lambda1), (cos((lambda1 - lambda2)) + 1.0));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(sin(lambda1), (cos((lambda1 - lambda2)) + 1.0d0))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin(lambda1), (Math.cos((lambda1 - lambda2)) + 1.0));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin(lambda1), (math.cos((lambda1 - lambda2)) + 1.0))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(lambda1), Float64(cos(Float64(lambda1 - lambda2)) + 1.0)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin(lambda1), (cos((lambda1 - lambda2)) + 1.0));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. lower--.f64 79.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Applied rewrites 79.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   2. lower-+.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

   3. lower-cos.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   4. lower--.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} + \cos \phi_1} \]

   5. lower-cos.f64 79.3

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\cos \phi_1}} \]

8. Applied rewrites 79.3%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}} \]

9. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

    2. lower-cos.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

    3. lower--.f64 70.6

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}} \]

11. Applied rewrites 70.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{1 + \cos \left(\lambda_1 - \lambda_2\right)}} \]

12. Taylor expanded in lambda2 around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
13. Step-by-step derivation

    1. lower-sin.f64 58.1

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

14. Applied rewrites 58.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

15. Final simplification 58.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \left(\lambda_1 - \lambda_2\right) + 1} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))