Midpoint on a great circle

Percentage Accurate: 98.7% → 99.6%

Time: 24.8s

Alternatives: 22

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-sum N/A

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

   4. cos-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 98.4

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

4. Applied egg-rr 98.4%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lower-*.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in phi1 around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. cos-neg N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 99.7

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

9. Simplified 99.7%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \lambda_1 \cdot \left(\sin \lambda_2 \cdot \mathsf{fma}\left(-0.16666666666666666, \lambda_1 \cdot \lambda_1, 1\right)\right)\right), \cos \phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (fma (sin (- lambda2)) (cos lambda1) (* (sin lambda1) (cos lambda2))))
   (fma
    (cos phi2)
    (fma
     (cos lambda1)
     (cos lambda2)
     (*
      lambda1
      (* (sin lambda2) (fma -0.16666666666666666 (* lambda1 lambda1) 1.0))))
    (cos phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * fma(sin(-lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2)))), fma(cos(phi2), fma(cos(lambda1), cos(lambda2), (lambda1 * (sin(lambda2) * fma(-0.16666666666666666, (lambda1 * lambda1), 1.0)))), cos(phi1)));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * fma(sin(Float64(-lambda2)), cos(lambda1), Float64(sin(lambda1) * cos(lambda2)))), fma(cos(phi2), fma(cos(lambda1), cos(lambda2), Float64(lambda1 * Float64(sin(lambda2) * fma(-0.16666666666666666, Float64(lambda1 * lambda1), 1.0)))), cos(phi1))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[(N[Sin[lambda2], $MachinePrecision] * N[(-0.16666666666666666 * N[(lambda1 * lambda1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-sum N/A

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

   4. cos-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 98.4

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

4. Applied egg-rr 98.4%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lower-*.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in phi1 around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. cos-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. cos-neg N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 99.7

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

9. Simplified 99.7%

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

10. Taylor expanded in lambda1 around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. distribute-rgt1-in N/A

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

    10. lower-*.f64 N/A

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

    11. lower-fma.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 N/A

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

    14. lower-sin.f64 98.8

        \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \lambda_1 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \lambda_1 \cdot \lambda_1, 1\right) \cdot \color{blue}{\sin \lambda_2}\right)\right), \cos \phi_1\right)} \]

12. Simplified 98.8%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\lambda_1 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \lambda_1 \cdot \lambda_1, 1\right) \cdot \sin \lambda_2\right)}\right), \cos \phi_1\right)} \]

13. Final simplification 98.8%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \cos \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \lambda_1 \cdot \left(\sin \lambda_2 \cdot \mathsf{fma}\left(-0.16666666666666666, \lambda_1 \cdot \lambda_1, 1\right)\right)\right), \cos \phi_1\right)} \]
14. Add Preprocessing

Alternative 3: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 + \cos \phi_2 \cdot t\_0\\ \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_2} \leq -3.14:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (+ (cos phi1) (* (cos phi2) t_0))))
   (if (<= (+ lambda1 (atan2 t_1 t_2)) -3.14)
     (+ lambda1 (atan2 (sin (- lambda2)) t_2))
     (+ lambda1 (atan2 t_1 (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos(phi1) + (cos(phi2) * t_0);
	double tmp;
	if ((lambda1 + atan2(t_1, t_2)) <= -3.14) {
		tmp = lambda1 + atan2(sin(-lambda2), t_2);
	} else {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    t_2 = cos(phi1) + (cos(phi2) * t_0)
    if ((lambda1 + atan2(t_1, t_2)) <= (-3.14d0)) then
        tmp = lambda1 + atan2(sin(-lambda2), t_2)
    else
        tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_2 = Math.cos(phi1) + (Math.cos(phi2) * t_0);
	double tmp;
	if ((lambda1 + Math.atan2(t_1, t_2)) <= -3.14) {
		tmp = lambda1 + Math.atan2(Math.sin(-lambda2), t_2);
	} else {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi1) + t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	t_2 = math.cos(phi1) + (math.cos(phi2) * t_0)
	tmp = 0
	if (lambda1 + math.atan2(t_1, t_2)) <= -3.14:
		tmp = lambda1 + math.atan2(math.sin(-lambda2), t_2)
	else:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi1) + t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi1) + Float64(cos(phi2) * t_0))
	tmp = 0.0
	if (Float64(lambda1 + atan(t_1, t_2)) <= -3.14)
		tmp = Float64(lambda1 + atan(sin(Float64(-lambda2)), t_2));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	t_2 = cos(phi1) + (cos(phi2) * t_0);
	tmp = 0.0;
	if ((lambda1 + atan2(t_1, t_2)) <= -3.14)
		tmp = lambda1 + atan2(sin(-lambda2), t_2);
	else
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 + N[ArcTan[t$95$1 / t$95$2], $MachinePrecision]), $MachinePrecision], -3.14], N[(lambda1 + N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

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)))))) < -3.14000000000000012

   1. Initial program 99.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-neg.f64 92.1

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

   8. Simplified 92.1%

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

   if -3.14000000000000012 < (+.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 97.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 76.5

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

   5. Simplified 76.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \leq -3.14:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-sum N/A

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

   4. cos-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 98.4

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

4. Applied egg-rr 98.4%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lift-sin.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lower-*.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in lambda2 around 0

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

   1. lower-sin.f64 98.5

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

9. Simplified 98.5%

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

Alternative 5: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-sum N/A

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

   4. cos-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 98.4

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

4. Applied egg-rr 98.4%

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

Alternative 6: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.9975:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1, \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1, \phi_1 \cdot 0.041666666666666664, -0.5\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.9975)
     (+ lambda1 (atan2 t_0 (fma (cos phi2) (cos lambda1) (cos phi1))))
     (+
      lambda1
      (atan2
       t_0
       (fma
        (cos phi2)
        (cos (- lambda1 lambda2))
        (fma
         (* phi1 phi1)
         (fma phi1 (* phi1 0.041666666666666664) -0.5)
         1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.9975) {
		tmp = lambda1 + atan2(t_0, fma(cos(phi2), cos(lambda1), cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, fma(cos(phi2), cos((lambda1 - lambda2)), fma((phi1 * phi1), fma(phi1, (phi1 * 0.041666666666666664), -0.5), 1.0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.9975)
		tmp = Float64(lambda1 + atan(t_0, fma(cos(phi2), cos(lambda1), cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, fma(cos(phi2), cos(Float64(lambda1 - lambda2)), fma(Float64(phi1 * phi1), fma(phi1, Float64(phi1 * 0.041666666666666664), -0.5), 1.0))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.9975], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * N[(phi1 * N[(phi1 * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99750000000000005

   1. Initial program 97.1%

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

   3. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 83.1

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

   5. Simplified 83.1%

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

   if 0.99750000000000005 < (cos.f64 phi1)

   1. Initial program 99.3%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 7: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.965:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(-0.5, \lambda_1 \cdot \lambda_1, 1\right), \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.965)
     (+
      lambda1
      (atan2
       t_0
       (fma (cos phi2) (fma -0.5 (* lambda1 lambda1) 1.0) (cos phi1))))
     (+ lambda1 (atan2 t_0 (fma (cos phi2) (cos (- lambda1 lambda2)) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.965) {
		tmp = lambda1 + atan2(t_0, fma(cos(phi2), fma(-0.5, (lambda1 * lambda1), 1.0), cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, fma(cos(phi2), cos((lambda1 - lambda2)), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.965)
		tmp = Float64(lambda1 + atan(t_0, fma(cos(phi2), fma(-0.5, Float64(lambda1 * lambda1), 1.0), cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, fma(cos(phi2), cos(Float64(lambda1 - lambda2)), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.965], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[(lambda1 * lambda1), $MachinePrecision] + 1.0), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.964999999999999969

   1. Initial program 97.0%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 95.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\lambda_1, \mathsf{fma}\left(\lambda_1, -0.5 \cdot \cos \lambda_2, \sin \left(-\lambda_2\right) \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot 0.16666666666666666, -1\right)\right), \cos \lambda_2\right)}} \]

   6. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 81.2

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

   8. Simplified 81.2%

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

   if 0.964999999999999969 < (cos.f64 phi1)

   1. Initial program 99.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 98.8

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

   5. Simplified 98.8%

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

Alternative 8: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.15:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (if (<= (cos phi2) 0.15)
     (+ lambda1 (atan2 (sin (- lambda2)) t_0))
     (+ lambda1 (atan2 (sin (- lambda1 lambda2)) t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)));
	double tmp;
	if (cos(phi2) <= 0.15) {
		tmp = lambda1 + atan2(sin(-lambda2), t_0);
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), t_0);
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))
    if (cos(phi2) <= 0.15d0) then
        tmp = lambda1 + atan2(sin(-lambda2), t_0)
    else
        tmp = lambda1 + atan2(sin((lambda1 - lambda2)), t_0)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)));
	double tmp;
	if (Math.cos(phi2) <= 0.15) {
		tmp = lambda1 + Math.atan2(Math.sin(-lambda2), t_0);
	} else {
		tmp = lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), t_0);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))
	tmp = 0
	if math.cos(phi2) <= 0.15:
		tmp = lambda1 + math.atan2(math.sin(-lambda2), t_0)
	else:
		tmp = lambda1 + math.atan2(math.sin((lambda1 - lambda2)), t_0)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (cos(phi2) <= 0.15)
		tmp = Float64(lambda1 + atan(sin(Float64(-lambda2)), t_0));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)));
	tmp = 0.0;
	if (cos(phi2) <= 0.15)
		tmp = lambda1 + atan2(sin(-lambda2), t_0);
	else
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.15], N[(lambda1 + N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.149999999999999994

   1. Initial program 99.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-neg.f64 59.0

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

   8. Simplified 59.0%

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

   if 0.149999999999999994 < (cos.f64 phi2)

   1. Initial program 98.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 87.2

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

   5. Simplified 87.2%

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

Alternative 9: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.31:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.31)
     (+ lambda1 (atan2 (sin (- lambda2)) (+ (cos phi1) (* (cos phi2) t_0))))
     (+ lambda1 (atan2 (sin (- lambda1 lambda2)) (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.31) {
		tmp = lambda1 + atan2(sin(-lambda2), (cos(phi1) + (cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (cos(phi2) <= 0.31d0) then
        tmp = lambda1 + atan2(sin(-lambda2), (cos(phi1) + (cos(phi2) * t_0)))
    else
        tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.31) {
		tmp = lambda1 + Math.atan2(Math.sin(-lambda2), (Math.cos(phi1) + (Math.cos(phi2) * t_0)));
	} else {
		tmp = lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(phi1) + t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.31:
		tmp = lambda1 + math.atan2(math.sin(-lambda2), (math.cos(phi1) + (math.cos(phi2) * t_0)))
	else:
		tmp = lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(phi1) + t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.31)
		tmp = Float64(lambda1 + atan(sin(Float64(-lambda2)), Float64(cos(phi1) + Float64(cos(phi2) * t_0))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.31)
		tmp = lambda1 + atan2(sin(-lambda2), (cos(phi1) + (cos(phi2) * t_0)));
	else
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + t_0));
	end
	tmp_2 = 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], 0.31], N[(lambda1 + N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.309999999999999998

   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

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-neg.f64 60.5

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

   8. Simplified 60.5%

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

   if 0.309999999999999998 < (cos.f64 phi2)

   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

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 88.0

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

   5. Simplified 88.0%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 88.0

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

   8. Simplified 88.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.31:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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]

Derivation

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. Add Preprocessing

Alternative 11: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in 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 97.2

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

5. Simplified 97.2%

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

Alternative 12: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{\mathsf{fma}\left(\cos \phi_2, t\_0, \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (sin (- lambda1 lambda2))))
   (if (<= phi2 8.5e-10)
     (+ lambda1 (atan2 t_1 (+ (cos phi1) t_0)))
     (+
      lambda1
      (atan2
       (* (cos phi2) t_1)
       (fma (cos phi2) t_0 (fma -0.5 (* phi1 phi1) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 8.5e-10) {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + t_0));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_1), fma(cos(phi2), t_0, fma(-0.5, (phi1 * phi1), 1.0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 8.5e-10)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + t_0)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), fma(cos(phi2), t_0, fma(-0.5, Float64(phi1 * phi1), 1.0))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 8.5e-10], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.4999999999999996e-10

   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

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 86.3

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

   5. Simplified 86.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 86.3

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

   8. Simplified 86.3%

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

   if 8.4999999999999996e-10 < phi2

   1. Initial program 98.1%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.3

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

   5. Simplified 82.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{\mathsf{fma}\left(\cos \phi_2, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (sin (- lambda1 lambda2))))
   (if (<= phi1 2.4e-7)
     (+ lambda1 (atan2 (* (cos phi2) t_1) (fma (cos phi2) t_0 1.0)))
     (+ lambda1 (atan2 t_1 (+ (cos phi1) (* (cos phi2) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 2.4e-7) {
		tmp = lambda1 + atan2((cos(phi2) * t_1), fma(cos(phi2), t_0, 1.0));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * t_0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= 2.4e-7)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_1), fma(cos(phi2), t_0, 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, 2.4e-7], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 2.39999999999999979e-7

   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 phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 83.2

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

   5. Simplified 83.2%

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

   if 2.39999999999999979e-7 < phi1

   1. Initial program 98.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 84.3

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

   5. Simplified 84.3%

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

Alternative 14: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 1450:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \left(\lambda_1 - \lambda_2\right) + \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \lambda_1 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi1 1450.0)
     (+
      lambda1
      (atan2
       t_0
       (+
        (cos (- lambda1 lambda2))
        (fma
         (* phi1 phi1)
         (fma (* phi1 phi1) 0.041666666666666664 -0.5)
         1.0))))
     (+ lambda1 (atan2 t_0 (+ (cos lambda1) (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 1450.0) {
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + fma((phi1 * phi1), fma((phi1 * phi1), 0.041666666666666664, -0.5), 1.0)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(lambda1) + cos(phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= 1450.0)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(Float64(lambda1 - lambda2)) + fma(Float64(phi1 * phi1), fma(Float64(phi1 * phi1), 0.041666666666666664, -0.5), 1.0))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(lambda1) + cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, 1450.0], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[lambda1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 1450

   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

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 75.1

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

   8. Simplified 75.1%

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

   9. Taylor expanded in phi1 around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. lower-+.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 67.6

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

   11. Simplified 67.6%

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

   if 1450 < phi1

   1. Initial program 98.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 85.1

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

   5. Simplified 85.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 83.6

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

   8. Simplified 83.6%

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

   9. Taylor expanded in lambda2 around 0

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

       1. lower-cos.f64 76.8

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

   11. Simplified 76.8%

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

Alternative 15: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 2.05 \cdot 10^{+45}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{t\_0 + \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 2.05e+45)
     (+
      lambda1
      (atan2 (sin (- lambda1 lambda2)) (+ t_0 (fma -0.5 (* phi1 phi1) 1.0))))
     (+ lambda1 (atan2 (sin lambda1) (+ (cos phi1) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 2.05e+45) {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (t_0 + fma(-0.5, (phi1 * phi1), 1.0)));
	} else {
		tmp = lambda1 + atan2(sin(lambda1), (cos(phi1) + t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= 2.05e+45)
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(t_0 + fma(-0.5, Float64(phi1 * phi1), 1.0))));
	else
		tmp = Float64(lambda1 + atan(sin(lambda1), Float64(cos(phi1) + t_0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, 2.05e+45], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 + N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 2.05000000000000006e45

   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

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 74.6

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

   8. Simplified 74.6%

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

   9. Taylor expanded in phi1 around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. lower-+.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 67.2

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

   11. Simplified 67.2%

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

   if 2.05000000000000006e45 < phi1

   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

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

      1. lower-sin.f64 N/A

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

      2. lower--.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 87.0

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

   8. Simplified 87.0%

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

   9. Taylor expanded in lambda2 around 0

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

       1. lower-sin.f64 77.7

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

   11. Simplified 77.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 2.05 \cdot 10^{+45}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 77.8

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

5. Simplified 77.8%

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

6. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 77.2

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

8. Simplified 77.2%

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

9. Final simplification 77.2%

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

Alternative 17: 76.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 77.8

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

5. Simplified 77.8%

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

6. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 77.2

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

8. Simplified 77.2%

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

9. Taylor expanded in lambda1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. cos-neg N/A

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

    4. lower-cos.f64 76.6

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

11. Simplified 76.6%

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

12. Final simplification 76.6%

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

Alternative 18: 67.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 77.8

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

5. Simplified 77.8%

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

6. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 77.2

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

8. Simplified 77.2%

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

9. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower--.f64 66.8

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

11. Simplified 66.8%

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

Alternative 19: 67.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 77.8

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

5. Simplified 77.8%

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

6. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 77.2

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

8. Simplified 77.2%

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

9. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower--.f64 66.8

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

11. Simplified 66.8%

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

12. Taylor expanded in lambda1 around 0

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

    1. cos-neg N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 N/A

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

    4. lower-cos.f64 66.7

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

14. Simplified 66.7%

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

Alternative 20: 62.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 77.8

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

5. Simplified 77.8%

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

6. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 77.2

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

8. Simplified 77.2%

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

9. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower--.f64 66.8

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

11. Simplified 66.8%

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

12. Taylor expanded in lambda2 around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 N/A

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

    3. lower-cos.f64 61.6

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

14. Simplified 61.6%

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

Alternative 21: 55.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. lower-sin.f64 N/A

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

   2. lower--.f64 77.8

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

5. Simplified 77.8%

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

6. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 77.2

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

8. Simplified 77.2%

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

9. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower--.f64 66.8

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

11. Simplified 66.8%

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

12. Taylor expanded in lambda2 around 0

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

    1. lower-sin.f64 54.1

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

14. Simplified 54.1%

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

Alternative 22: 52.8% accurate, 27.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\lambda_1}} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (/ 1.0 (/ 1.0 lambda1)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return 1.0 / (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 = 1.0d0 / (1.0d0 / lambda1)
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return 1.0 / (1.0 / lambda1);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return 1.0 / (1.0 / lambda1)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(1.0 / Float64(1.0 / lambda1))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = 1.0 / (1.0 / lambda1);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(1.0 / N[(1.0 / lambda1), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-atan2.f64 N/A

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in lambda1 around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\lambda_1}}} \]
6. Step-by-step derivation

   1. lower-/.f64 49.0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\lambda_1}}} \]

7. Simplified 49.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\lambda_1}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024210 
(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)))))))