Midpoint on a great circle

Percentage Accurate: 98.7% → 99.7%

Time: 22.9s

Alternatives: 20

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: 98.7% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (fma (sin (- lambda2)) (cos lambda1) (* (cos lambda2) (sin lambda1))))
   (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), (cos(lambda2) * sin(lambda1)))), 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(cos(lambda2) * sin(lambda1)))), 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[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $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 98.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift--.f64 N/A

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

   6. cos-diff N/A

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

   7. distribute-lft-in N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

4. Applied rewrites 98.3%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. sin-sum N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-neg N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in phi2 around inf

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

   1. +-commutative N/A

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

   2. cos-neg N/A

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

   3. associate-*r* N/A

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

   4. cos-neg N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-fma.f64 N/A

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

9. Applied rewrites 99.6%

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

Alternative 2: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\left(\phi_1 \cdot \phi_1\right) \cdot -0.5}\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot t\_0}\\ \mathbf{if}\;t\_3 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(0.16666666666666666, \lambda_2 \cdot \lambda_2, -1\right), \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_2 \cdot \lambda_2, 0.08333333333333333, -0.16666666666666666\right), \lambda_2 \cdot \mathsf{fma}\left(\lambda_2 \cdot \lambda_2, -0.08333333333333333, 0.5\right)\right), -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right), \lambda_1\right)\right)}{\cos \phi_1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (+ lambda1 (atan2 t_1 (* (* phi1 phi1) -0.5))))
        (t_3 (+ lambda1 (atan2 t_1 (+ (cos phi1) (* (cos phi2) t_0))))))
   (if (<= t_3 -0.01)
     t_2
     (if (<= t_3 2e-6)
       (+
        lambda1
        (atan2
         (fma
          lambda2
          (fma 0.16666666666666666 (* lambda2 lambda2) -1.0)
          (fma
           lambda1
           (fma
            lambda1
            (fma
             lambda1
             (fma (* lambda2 lambda2) 0.08333333333333333 -0.16666666666666666)
             (* lambda2 (fma (* lambda2 lambda2) -0.08333333333333333 0.5)))
            (* -0.5 (* lambda2 lambda2)))
           lambda1))
         (+ (cos phi1) t_0)))
       t_2))))

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 t_2 = lambda1 + atan2(t_1, ((phi1 * phi1) * -0.5));
	double t_3 = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * t_0)));
	double tmp;
	if (t_3 <= -0.01) {
		tmp = t_2;
	} else if (t_3 <= 2e-6) {
		tmp = lambda1 + atan2(fma(lambda2, fma(0.16666666666666666, (lambda2 * lambda2), -1.0), fma(lambda1, fma(lambda1, fma(lambda1, fma((lambda2 * lambda2), 0.08333333333333333, -0.16666666666666666), (lambda2 * fma((lambda2 * lambda2), -0.08333333333333333, 0.5))), (-0.5 * (lambda2 * lambda2))), lambda1)), (cos(phi1) + t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = Float64(lambda1 + atan(t_1, Float64(Float64(phi1 * phi1) * -0.5)))
	t_3 = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * t_0))))
	tmp = 0.0
	if (t_3 <= -0.01)
		tmp = t_2;
	elseif (t_3 <= 2e-6)
		tmp = Float64(lambda1 + atan(fma(lambda2, fma(0.16666666666666666, Float64(lambda2 * lambda2), -1.0), fma(lambda1, fma(lambda1, fma(lambda1, fma(Float64(lambda2 * lambda2), 0.08333333333333333, -0.16666666666666666), Float64(lambda2 * fma(Float64(lambda2 * lambda2), -0.08333333333333333, 0.5))), Float64(-0.5 * Float64(lambda2 * lambda2))), lambda1)), Float64(cos(phi1) + t_0)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(phi1 * phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.01], t$95$2, If[LessEqual[t$95$3, 2e-6], N[(lambda1 + N[ArcTan[N[(lambda2 * N[(0.16666666666666666 * N[(lambda2 * lambda2), $MachinePrecision] + -1.0), $MachinePrecision] + N[(lambda1 * N[(lambda1 * N[(lambda1 * N[(N[(lambda2 * lambda2), $MachinePrecision] * 0.08333333333333333 + -0.16666666666666666), $MachinePrecision] + N[(lambda2 * N[(N[(lambda2 * lambda2), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(lambda2 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + lambda1), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 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)))))) < -0.0100000000000000002 or 1.99999999999999991e-6 < (+.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 98.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 85.3

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in phi1 around inf

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

   10. Applied rewrites 72.2%

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

   if -0.0100000000000000002 < (+.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)))))) < 1.99999999999999991e-6

   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 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 61.9

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

   5. Applied rewrites 61.9%

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

   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. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1 + \color{blue}{\lambda_2 \cdot \left(-1 \cdot \cos \lambda_1 + \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)}}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.1%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\lambda_2, \color{blue}{\mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(\sin \lambda_1, -0.5, 0.16666666666666666 \cdot \left(\cos \lambda_1 \cdot \lambda_2\right)\right), -\cos \lambda_1\right)}, \sin \lambda_1\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \left(1 + \left(\frac{-1}{2} \cdot {\lambda_2}^{2} + \lambda_1 \cdot \left(\lambda_1 \cdot \left(\frac{1}{12} \cdot {\lambda_2}^{2} - \frac{1}{6}\right) + \lambda_2 \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {\lambda_2}^{2}\right)\right)\right)\right) + \lambda_2 \cdot \color{blue}{\left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)}}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 62.1%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(0.16666666666666666, \color{blue}{\lambda_2 \cdot \lambda_2}, -1\right), \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_2 \cdot \lambda_2, 0.08333333333333333, -0.16666666666666666\right), \lambda_2 \cdot \mathsf{fma}\left(\lambda_2 \cdot \lambda_2, -0.08333333333333333, 0.5\right)\right), -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right), \lambda_1\right)\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \leq -0.01:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(\phi_1 \cdot \phi_1\right) \cdot -0.5}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(0.16666666666666666, \lambda_2 \cdot \lambda_2, -1\right), \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_2 \cdot \lambda_2, 0.08333333333333333, -0.16666666666666666\right), \lambda_2 \cdot \mathsf{fma}\left(\lambda_2 \cdot \lambda_2, -0.08333333333333333, 0.5\right)\right), -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right), \lambda_1\right)\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(\phi_1 \cdot \phi_1\right) \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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)))))))