Midpoint on a great circle

Percentage Accurate: 98.7% → 99.6%

Time: 14.4s

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: 98.7% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. cancel-sub-sign-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in lambda1 around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 75.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + t\_0} + \lambda_1\\ t_3 := \tan^{-1}_* \frac{t\_1}{t\_0 \cdot \cos \phi_2 + \cos \phi_1} + \lambda_1\\ \mathbf{if}\;t\_3 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.0002:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \lambda_1 + \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (+ (atan2 t_1 (+ (fma (* phi1 phi1) -0.5 1.0) t_0)) lambda1))
        (t_3 (+ (atan2 t_1 (+ (* t_0 (cos phi2)) (cos phi1))) lambda1)))
   (if (<= t_3 -0.05)
     t_2
     (if (<= t_3 0.0002)
       (+ (atan2 t_1 (+ (cos lambda1) (cos phi1))) lambda1)
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double t_2 = atan2(t_1, (fma((phi1 * phi1), -0.5, 1.0) + t_0)) + lambda1;
	double t_3 = atan2(t_1, ((t_0 * cos(phi2)) + cos(phi1))) + lambda1;
	double tmp;
	if (t_3 <= -0.05) {
		tmp = t_2;
	} else if (t_3 <= 0.0002) {
		tmp = atan2(t_1, (cos(lambda1) + cos(phi1))) + lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	t_2 = Float64(atan(t_1, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + t_0)) + lambda1)
	t_3 = Float64(atan(t_1, Float64(Float64(t_0 * cos(phi2)) + cos(phi1))) + lambda1)
	tmp = 0.0
	if (t_3 <= -0.05)
		tmp = t_2;
	elseif (t_3 <= 0.0002)
		tmp = Float64(atan(t_1, Float64(cos(lambda1) + cos(phi1))) + lambda1);
	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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[t$95$1 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[t$95$1 / N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[t$95$3, -0.05], t$95$2, If[LessEqual[t$95$3, 0.0002], N[(N[ArcTan[t$95$1 / N[(N[Cos[lambda1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $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)))))) < -0.050000000000000003 or 2.0000000000000001e-4 < (+.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 98.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 80.0%

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

   if -0.050000000000000003 < (+.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)))))) < 2.0000000000000001e-4

   1. Initial program 99.2%

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

   3. 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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 57.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_1} + \lambda_1 \leq -0.05:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \mathbf{elif}\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_1} + \lambda_1 \leq 0.0002:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \lambda_1 + \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  (atan2
   (*
    (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1))))
    (cos phi2))
   (+
    (*
     (fma
      (fma (* (cos lambda2) lambda1) -0.5 (sin lambda2))
      lambda1
      (cos lambda2))
     (cos phi2))
    (cos phi1)))
  lambda1))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1))) * cos(phi2)), ((fma(fma((cos(lambda2) * lambda1), -0.5, sin(lambda2)), lambda1, cos(lambda2)) * cos(phi2)) + cos(phi1))) + lambda1;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1)))) * cos(phi2)), Float64(Float64(fma(fma(Float64(cos(lambda2) * lambda1), -0.5, sin(lambda2)), lambda1, cos(lambda2)) * cos(phi2)) + cos(phi1))) + lambda1)
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * lambda1), $MachinePrecision] * -0.5 + N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * lambda1 + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. cancel-sub-sign-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in lambda1 around 0

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

   1. cos-neg-rev N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. cos-neg-rev N/A

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

   5. sin-neg N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 99.0%

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

8. Final simplification 99.0%

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

Alternative 4: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-cos.f64 N/A

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

   6. sin-neg N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-sin.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in lambda1 around 0

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

   1. cos-neg-rev N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

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

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

   6. sin-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.9999:\\ \;\;\;\;\tan^{-1}_* \frac{\left(-\cos \phi_2\right) \cdot \sin \lambda_2}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_1 \cdot \phi_1, -0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi1) 0.9999)
   (+
    (atan2
     (* (- (cos phi2)) (sin lambda2))
     (fma (cos lambda2) (cos phi2) (cos phi1)))
    lambda1)
   (+
    (atan2
     (* (sin (- lambda1 lambda2)) (cos phi2))
     (fma
      (* (fma 0.041666666666666664 (* phi1 phi1) -0.5) phi1)
      phi1
      (fma (cos (- lambda2 lambda1)) (cos phi2) 1.0)))
    lambda1)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi1) <= 0.9999) {
		tmp = atan2((-cos(phi2) * sin(lambda2)), fma(cos(lambda2), cos(phi2), cos(phi1))) + lambda1;
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), fma((fma(0.041666666666666664, (phi1 * phi1), -0.5) * phi1), phi1, fma(cos((lambda2 - lambda1)), cos(phi2), 1.0))) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi1) <= 0.9999)
		tmp = Float64(atan(Float64(Float64(-cos(phi2)) * sin(lambda2)), fma(cos(lambda2), cos(phi2), cos(phi1))) + lambda1);
	else
		tmp = Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(Float64(fma(0.041666666666666664, Float64(phi1 * phi1), -0.5) * phi1), phi1, fma(cos(Float64(lambda2 - lambda1)), cos(phi2), 1.0))) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.9999], N[(N[ArcTan[N[((-N[Cos[phi2], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.041666666666666664 * N[(phi1 * phi1), $MachinePrecision] + -0.5), $MachinePrecision] * phi1), $MachinePrecision] * phi1 + N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99990000000000001

   1. Initial program 97.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. cancel-sub-sign-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. cos-neg-rev N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 88.5

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

   10. Applied rewrites 88.5%

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

   if 0.99990000000000001 < (cos.f64 phi1)

   1. Initial program 99.2%

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

   3. Taylor expanded in phi1 around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_1 \cdot \phi_1, -0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.9999:\\ \;\;\;\;\tan^{-1}_* \frac{\left(-\cos \phi_2\right) \cdot \sin \lambda_2}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \phi_1 \cdot \phi_1, -0.5\right) \cdot \phi_1, \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_2 \leq 0.986:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-0.5 \cdot \phi_1, \phi_1, \mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (cos phi2) 0.986)
     (+ (atan2 t_1 (fma (* -0.5 phi1) phi1 (fma t_0 (cos phi2) 1.0))) lambda1)
     (+ (atan2 t_1 (+ t_0 (cos phi1))) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (cos(phi2) <= 0.986) {
		tmp = atan2(t_1, fma((-0.5 * phi1), phi1, fma(t_0, cos(phi2), 1.0))) + lambda1;
	} else {
		tmp = atan2(t_1, (t_0 + cos(phi1))) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (cos(phi2) <= 0.986)
		tmp = Float64(atan(t_1, fma(Float64(-0.5 * phi1), phi1, fma(t_0, cos(phi2), 1.0))) + lambda1);
	else
		tmp = Float64(atan(t_1, Float64(t_0 + cos(phi1))) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.986], N[(N[ArcTan[t$95$1 / N[(N[(-0.5 * phi1), $MachinePrecision] * phi1 + N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$1 / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.98599999999999999

   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 phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-in N/A

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

      17. +-commutative N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 N/A

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

      23. lower-cos.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if 0.98599999999999999 < (cos.f64 phi2)

   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. 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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.986:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-0.5 \cdot \phi_1, \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_2 - \lambda_1\right) + \cos \phi_1} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_1 \leq 0.992:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right), \cos \lambda_2, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (cos phi1) 0.992)
     (+
      (atan2 t_0 (fma (fma (* phi2 phi2) -0.5 1.0) (cos lambda2) (cos phi1)))
      lambda1)
     (+ (atan2 t_0 (fma (cos (- lambda2 lambda1)) (cos phi2) 1.0)) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (cos(phi1) <= 0.992) {
		tmp = atan2(t_0, fma(fma((phi2 * phi2), -0.5, 1.0), cos(lambda2), cos(phi1))) + lambda1;
	} else {
		tmp = atan2(t_0, fma(cos((lambda2 - lambda1)), cos(phi2), 1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (cos(phi1) <= 0.992)
		tmp = Float64(atan(t_0, fma(fma(Float64(phi2 * phi2), -0.5, 1.0), cos(lambda2), cos(phi1))) + lambda1);
	else
		tmp = Float64(atan(t_0, fma(cos(Float64(lambda2 - lambda1)), cos(phi2), 1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.992], N[(N[ArcTan[t$95$0 / N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99199999999999999

   1. Initial program 97.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 + \left(\cos \left(\lambda_1 - \lambda_2\right) + \frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 N/A

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

      19. lower-cos.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 76.8%

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

   if 0.99199999999999999 < (cos.f64 phi1)

   1. Initial program 99.2%

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

   3. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-cos.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.992:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right), \cos \lambda_2, \cos \phi_1\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_1 \leq 0.992:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(t\_0, \cos \phi_2, 1\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (cos phi1) 0.992)
     (+ (atan2 t_1 (+ t_0 (cos phi1))) lambda1)
     (+ (atan2 t_1 (fma t_0 (cos phi2) 1.0)) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (cos(phi1) <= 0.992) {
		tmp = atan2(t_1, (t_0 + cos(phi1))) + lambda1;
	} else {
		tmp = atan2(t_1, fma(t_0, cos(phi2), 1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (cos(phi1) <= 0.992)
		tmp = Float64(atan(t_1, Float64(t_0 + cos(phi1))) + lambda1);
	else
		tmp = Float64(atan(t_1, fma(t_0, cos(phi2), 1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.992], N[(N[ArcTan[t$95$1 / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$1 / N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99199999999999999

   1. Initial program 97.9%

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

   3. Taylor expanded in phi2 around 0

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

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if 0.99199999999999999 < (cos.f64 phi1)

   1. Initial program 99.2%

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

   3. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. cos-neg N/A

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

      10. lower-cos.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-cos.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.992:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_2 - \lambda_1\right) + \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2, 1\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \left(\lambda_2 - \lambda_1\right) + \cos \phi_1} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (cos phi2) -0.1)
     (+
      (atan2 t_0 (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda1 lambda2))))
      lambda1)
     (+ (atan2 t_0 (+ (cos (- lambda2 lambda1)) (cos phi1))) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (cos(phi2) <= -0.1) {
		tmp = atan2(t_0, (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda1 - lambda2)))) + lambda1;
	} else {
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) + cos(phi1))) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (cos(phi2) <= -0.1)
		tmp = Float64(atan(t_0, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda1 - lambda2)))) + lambda1);
	else
		tmp = Float64(atan(t_0, Float64(cos(Float64(lambda2 - lambda1)) + cos(phi1))) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.1], N[(N[ArcTan[t$95$0 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 65.1%

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

   if -0.10000000000000001 < (cos.f64 phi2)

   1. Initial program 98.3%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 83.5

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

   5. Applied rewrites 83.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_2 - \lambda_1\right) + \cos \phi_1} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Final simplification 98.6%

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

Alternative 11: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \lambda_2 + \cos \phi_1} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= (cos phi2) -0.1)
     (+
      (atan2 t_0 (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda1 lambda2))))
      lambda1)
     (+ (atan2 t_0 (+ (cos lambda2) (cos phi1))) lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (cos(phi2) <= -0.1) {
		tmp = atan2(t_0, (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda1 - lambda2)))) + lambda1;
	} else {
		tmp = atan2(t_0, (cos(lambda2) + cos(phi1))) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (cos(phi2) <= -0.1)
		tmp = Float64(atan(t_0, Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda1 - lambda2)))) + lambda1);
	else
		tmp = Float64(atan(t_0, Float64(cos(lambda2) + cos(phi1))) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.1], N[(N[ArcTan[t$95$0 / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$0 / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 65.1%

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

   if -0.10000000000000001 < (cos.f64 phi2)

   1. Initial program 98.3%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 83.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.1:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \lambda_2 + \cos \phi_1} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
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 98.4

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 13: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.102:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + t\_0} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{1 + t\_0} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (cos phi2) -0.102)
     (+
      (atan2 (* (sin lambda1) (cos phi2)) (+ (fma (* phi1 phi1) -0.5 1.0) t_0))
      lambda1)
     (+
      (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (+ 1.0 t_0))
      lambda1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= -0.102) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (fma((phi1 * phi1), -0.5, 1.0) + t_0)) + lambda1;
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (1.0 + t_0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= -0.102)
		tmp = Float64(atan(Float64(sin(lambda1) * cos(phi2)), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + t_0)) + lambda1);
	else
		tmp = Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(1.0 + t_0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], -0.102], N[(N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < -0.101999999999999993

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0

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

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in lambda2 around 0

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

       1. lower-sin.f64 49.3

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

   11. Applied rewrites 49.3%

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

   12. Taylor expanded in phi1 around 0

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

   14. Applied rewrites 62.5%

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

   if -0.101999999999999993 < (cos.f64 phi2)

   1. Initial program 98.3%

      \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq -0.102:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{1 + \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  (atan2
   (* (sin (- lambda1 lambda2)) (cos phi2))
   (+ (fma (* phi1 phi1) -0.5 1.0) (cos (- lambda1 lambda2))))
  lambda1))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), (fma((phi1 * phi1), -0.5, 1.0) + cos((lambda1 - lambda2)))) + lambda1;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(fma(Float64(phi1 * phi1), -0.5, 1.0) + cos(Float64(lambda1 - lambda2)))) + lambda1)
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
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. sub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-cos.f64 76.6

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 68.5%

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

9. Final simplification 68.5%

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

Alternative 15: 68.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (1.0d0 + cos((lambda1 - lambda2)))) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
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. sub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-cos.f64 76.6

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 67.2%

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

9. Final simplification 67.2%

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

Alternative 16: 68.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (cos(lambda2) + 1.0d0)) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
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. sub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-cos.f64 76.6

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 67.2%

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

9. Taylor expanded in lambda1 around 0

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

11. Applied rewrites 67.2%

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

12. Final simplification 67.2%

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

Alternative 17: 63.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (cos(lambda1) + 1.0d0)) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
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. sub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-cos.f64 76.6

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 67.2%

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

9. Taylor expanded in lambda2 around 0

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

11. Applied rewrites 61.9%

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

12. Final simplification 61.9%

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

Alternative 18: 55.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \lambda_1 + 1} + \lambda_1 \end{array} \]

FPCore

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

C

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

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 = atan2((sin(lambda1) * cos(phi2)), (cos(lambda1) + 1.0d0)) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
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. sub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-cos.f64 76.6

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 67.2%

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

9. Taylor expanded in lambda2 around 0

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

    1. lower-sin.f64 53.6

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

11. Applied rewrites 53.6%

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

12. Taylor expanded in lambda2 around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{1 + \cos \lambda_1} \]
13. Step-by-step derivation

14. Applied rewrites 53.6%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{1 + \cos \lambda_1} \]

15. Final simplification 53.6%

    \[\leadsto \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \lambda_1 + 1} + \lambda_1 \]
16. Add Preprocessing

Reproduce

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