Bearing on a great circle

Percentage Accurate: 79.6% → 99.7%

Time: 31.1s

Alternatives: 34

Speedup: 1.0×

• Could not converge on a ground truth

  1. lambda1 = 9.06411491081843e+65
  2. lambda2 = -6.31410545222873e+65
  3. phi1 = 5.696830491450221e-283
  4. phi2 = 2.3313671962105135e+213

Specification

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. cos-lowering-cos.f64 79.2

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

4. Applied egg-rr 79.2%

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. sin-sum N/A

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

   4. cos-neg N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sin-neg N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. sin-lowering-sin.f64 N/A

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

   14. cos-lowering-cos.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in lambda1 around 0

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. sin-diff N/A

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

   2. sub-neg N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. sin-neg N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. neg-lowering-neg.f64 88.7

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

4. Applied egg-rr 88.7%

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

   1. cos-diff N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. sin-lowering-sin.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. cos-lowering-cos.f64 79.2

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

4. Applied egg-rr 79.2%

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

   1. sin-diff N/A

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

   2. --lowering--.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sin-lowering-sin.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \sin \left(-\lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.017:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(t\_3, \cos \lambda_1, t\_0\right)}{\mathsf{fma}\left(\sin \phi_1, t\_1 \cdot \left(-\cos \phi_2\right), t\_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot t\_3\right)}{t\_2 - \sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0 - \left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \cos \lambda_1}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(-t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda2) (sin lambda1)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi1) (sin phi2)))
        (t_3 (sin (- lambda2))))
   (if (<= phi2 -0.017)
     (atan2
      (* (cos phi2) (fma t_3 (cos lambda1) t_0))
      (fma (sin phi1) (* t_1 (- (cos phi2))) t_2))
     (if (<= phi2 2e-26)
       (atan2
        (* (cos phi2) (fma (sin lambda1) (cos lambda2) (* (cos lambda1) t_3)))
        (-
         t_2
         (*
          (sin phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (atan2
        (- (* (cos phi2) t_0) (* (* (cos phi2) (sin lambda2)) (cos lambda1)))
        (fma (sin phi2) (cos phi1) (* (* (cos phi2) (sin phi1)) (- t_1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda2) * sin(lambda1);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi1) * sin(phi2);
	double t_3 = sin(-lambda2);
	double tmp;
	if (phi2 <= -0.017) {
		tmp = atan2((cos(phi2) * fma(t_3, cos(lambda1), t_0)), fma(sin(phi1), (t_1 * -cos(phi2)), t_2));
	} else if (phi2 <= 2e-26) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * t_3))), (t_2 - (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(((cos(phi2) * t_0) - ((cos(phi2) * sin(lambda2)) * cos(lambda1))), fma(sin(phi2), cos(phi1), ((cos(phi2) * sin(phi1)) * -t_1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda2) * sin(lambda1))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi1) * sin(phi2))
	t_3 = sin(Float64(-lambda2))
	tmp = 0.0
	if (phi2 <= -0.017)
		tmp = atan(Float64(cos(phi2) * fma(t_3, cos(lambda1), t_0)), fma(sin(phi1), Float64(t_1 * Float64(-cos(phi2))), t_2));
	elseif (phi2 <= 2e-26)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * t_3))), Float64(t_2 - Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(Float64(Float64(cos(phi2) * t_0) - Float64(Float64(cos(phi2) * sin(lambda2)) * cos(lambda1))), fma(sin(phi2), cos(phi1), Float64(Float64(cos(phi2) * sin(phi1)) * Float64(-t_1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[(-lambda2)], $MachinePrecision]}, If[LessEqual[phi2, -0.017], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$3 * N[Cos[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * N[(t$95$1 * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2e-26], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-t$95$1)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.017000000000000001

   1. Initial program 68.8%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 68.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sin-sum N/A

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

      4. cos-neg N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. cos-lowering-cos.f64 83.9

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

   6. Applied egg-rr 83.9%

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

   if -0.017000000000000001 < phi2 < 2.0000000000000001e-26

   1. Initial program 85.7%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 91.8

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

   4. Applied egg-rr 91.8%

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

      1. cos-diff N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. sin-lowering-sin.f64 99.3

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

   9. Simplified 99.3%

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

   if 2.0000000000000001e-26 < phi2

   1. Initial program 77.6%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 77.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. *-lowering-*.f64 N/A

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

      16. cos-lowering-cos.f64 N/A

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

      17. neg-lowering-neg.f64 N/A

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

      18. sin-lowering-sin.f64 77.6

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

   6. Applied egg-rr 77.6%

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

      1. *-commutative N/A

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

      2. sin-diff N/A

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

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

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

      4. distribute-rgt-in N/A

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

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

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

      6. distribute-lft-neg-out N/A

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

      7. unsub-neg N/A

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

      8. --lowering--.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sin-lowering-sin.f64 N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. associate-*l* N/A

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

   8. Applied egg-rr 88.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.017:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \left(-\lambda_2\right), \cos \lambda_1, \cos \lambda_2 \cdot \sin \lambda_1\right)}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \phi_2 \cdot \sin \lambda_2\right) \cdot \cos \lambda_1}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \cos \lambda_1, \sin \phi_1, t\_0\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.052:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \lambda_1 - \sin \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1, \left(-\cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right), t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (atan2
          (*
           (cos phi2)
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (cos lambda1) (sin (- lambda2)))))
          (fma (- (* (cos phi2) (cos lambda1))) (sin phi1) t_0))))
   (if (<= lambda1 -0.052)
     t_1
     (if (<= lambda1 1.02e-8)
       (atan2
        (* (cos phi2) (- (* (cos lambda2) lambda1) (sin lambda2)))
        (fma
         (sin phi1)
         (* (- (cos phi2)) (fma lambda1 (sin lambda2) (cos lambda2)))
         t_0))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), fma(-(cos(phi2) * cos(lambda1)), sin(phi1), t_0));
	double tmp;
	if (lambda1 <= -0.052) {
		tmp = t_1;
	} else if (lambda1 <= 1.02e-8) {
		tmp = atan2((cos(phi2) * ((cos(lambda2) * lambda1) - sin(lambda2))), fma(sin(phi1), (-cos(phi2) * fma(lambda1, sin(lambda2), cos(lambda2))), t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), fma(Float64(-Float64(cos(phi2) * cos(lambda1))), sin(phi1), t_0))
	tmp = 0.0
	if (lambda1 <= -0.052)
		tmp = t_1;
	elseif (lambda1 <= 1.02e-8)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * lambda1) - sin(lambda2))), fma(sin(phi1), Float64(Float64(-cos(phi2)) * fma(lambda1, sin(lambda2), cos(lambda2))), t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]) * N[Sin[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -0.052], t$95$1, If[LessEqual[lambda1, 1.02e-8], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * lambda1), $MachinePrecision] - N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * N[((-N[Cos[phi2], $MachinePrecision]) * N[(lambda1 * N[Sin[lambda2], $MachinePrecision] + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.0519999999999999976 or 1.02000000000000003e-8 < lambda1

   1. Initial program 60.9%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 79.7

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

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in lambda2 around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. cos-lowering-cos.f64 N/A

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

      16. sin-lowering-sin.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. cos-lowering-cos.f64 N/A

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

      19. sin-lowering-sin.f64 79.6

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

   7. Simplified 79.6%

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

   if -0.0519999999999999976 < lambda1 < 1.02000000000000003e-8

   1. Initial program 97.5%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. sin-neg N/A

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

      5. remove-double-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. cos-lowering-cos.f64 97.7

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

   7. Simplified 97.7%

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

   8. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. cos-neg N/A

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

      3. sin-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. sin-lowering-sin.f64 98.9

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

   10. Simplified 98.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.052:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \cos \lambda_1, \sin \phi_1, \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \lambda_1 - \sin \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1, \left(-\cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \cos \lambda_1, \sin \phi_1, \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \left(-\sin \phi_1\right)\right)\right)}\\ \mathbf{if}\;\lambda_2 \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \sin \lambda_1, \left(-\lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (cos phi2)
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (cos lambda1) (sin (- lambda2)))))
          (fma
           (cos phi1)
           (sin phi2)
           (* (cos phi2) (* (cos lambda2) (- (sin phi1))))))))
   (if (<= lambda2 -7.5e-6)
     t_0
     (if (<= lambda2 7.5e-14)
       (atan2
        (fma
         (cos phi2)
         (sin lambda1)
         (* (- lambda2) (* (cos phi2) (cos lambda1))))
        (fma
         (sin phi1)
         (* (cos (- lambda1 lambda2)) (- (cos phi2)))
         (* (cos phi1) (sin phi2))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), fma(cos(phi1), sin(phi2), (cos(phi2) * (cos(lambda2) * -sin(phi1)))));
	double tmp;
	if (lambda2 <= -7.5e-6) {
		tmp = t_0;
	} else if (lambda2 <= 7.5e-14) {
		tmp = atan2(fma(cos(phi2), sin(lambda1), (-lambda2 * (cos(phi2) * cos(lambda1)))), fma(sin(phi1), (cos((lambda1 - lambda2)) * -cos(phi2)), (cos(phi1) * sin(phi2))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), fma(cos(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(lambda2) * Float64(-sin(phi1))))))
	tmp = 0.0
	if (lambda2 <= -7.5e-6)
		tmp = t_0;
	elseif (lambda2 <= 7.5e-14)
		tmp = atan(fma(cos(phi2), sin(lambda1), Float64(Float64(-lambda2) * Float64(cos(phi2) * cos(lambda1)))), fma(sin(phi1), Float64(cos(Float64(lambda1 - lambda2)) * Float64(-cos(phi2))), Float64(cos(phi1) * sin(phi2))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -7.5e-6], t$95$0, If[LessEqual[lambda2, 7.5e-14], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[((-lambda2) * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -7.50000000000000019e-6 or 7.4999999999999996e-14 < lambda2

   1. Initial program 63.1%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 80.2

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

   4. Applied egg-rr 80.2%

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

   5. Taylor expanded in lambda1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sin-lowering-sin.f64 N/A

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

      13. cos-neg N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. cos-lowering-cos.f64 79.8

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

   7. Simplified 79.8%

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

   if -7.50000000000000019e-6 < lambda2 < 7.4999999999999996e-14

   1. Initial program 99.6%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. cos-lowering-cos.f64 99.8

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

   7. Simplified 99.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \left(-\sin \phi_1\right)\right)\right)}\\ \mathbf{elif}\;\lambda_2 \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \phi_2, \sin \lambda_1, \left(-\lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \left(-\sin \phi_1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. atan2-lowering-atan2.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sin-lowering-sin.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 79.1%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-sum N/A

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

   4. cos-neg N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. neg-lowering-neg.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sin-lowering-sin.f64 N/A

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

   12. cos-lowering-cos.f64 88.7

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

6. Applied egg-rr 88.7%

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

7. Final simplification 88.7%

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

Alternative 8: 89.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. sin-diff N/A

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

   2. sub-neg N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. sin-neg N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. sin-lowering-sin.f64 N/A

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

   11. neg-lowering-neg.f64 88.7

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

4. Applied egg-rr 88.7%

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

5. Final simplification 88.7%

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

Alternative 9: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, t\_0\right)}{t\_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t\_2}\\ \mathbf{if}\;\phi_1 \leq -120:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 5600:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot t\_0\right)}{t\_1 - \sin \phi_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3
         (atan2
          (* (cos phi2) (fma (sin lambda1) (cos lambda2) t_0))
          (- t_1 (* (* (cos phi2) (sin phi1)) t_2)))))
   (if (<= phi1 -120.0)
     t_3
     (if (<= phi1 5600.0)
       (atan2
        (* (cos phi2) (fma (sin lambda1) (cos lambda2) (* (cos lambda1) t_0)))
        (- t_1 (* (sin phi1) t_2)))
       t_3))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(-lambda2);
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * t_2)));
	double tmp;
	if (phi1 <= -120.0) {
		tmp = t_3;
	} else if (phi1 <= 5600.0) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * t_0))), (t_1 - (sin(phi1) * t_2)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), t_0)), Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * t_2)))
	tmp = 0.0
	if (phi1 <= -120.0)
		tmp = t_3;
	elseif (phi1 <= 5600.0)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * t_0))), Float64(t_1 - Float64(sin(phi1) * t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[(-lambda2)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -120.0], t$95$3, If[LessEqual[phi1, 5600.0], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -120 or 5600 < phi1

   1. Initial program 77.6%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 80.0

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in lambda1 around 0

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

      1. sin-lowering-sin.f64 N/A

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

      2. neg-lowering-neg.f64 78.6

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

   7. Simplified 78.6%

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

   if -120 < phi1 < 5600

   1. Initial program 80.6%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. sin-lowering-sin.f64 97.4

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

   7. Simplified 97.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -120:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \left(-\lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 5600:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \left(-\lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, t\_0\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot t\_0\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \phi_1, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda2)))
        (t_1
         (atan2
          (* (cos phi2) (fma (sin lambda1) (cos lambda2) t_0))
          (-
           (* (cos phi1) (sin phi2))
           (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))))
   (if (<= phi1 -7.2e-6)
     t_1
     (if (<= phi1 2.4e-5)
       (atan2
        (* (cos phi2) (fma (sin lambda1) (cos lambda2) (* (cos lambda1) t_0)))
        (fma (- (* (cos phi2) phi1)) (cos (- lambda2 lambda1)) (sin phi2)))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(-lambda2);
	double t_1 = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), t_0)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi1 <= -7.2e-6) {
		tmp = t_1;
	} else if (phi1 <= 2.4e-5) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * t_0))), fma(-(cos(phi2) * phi1), cos((lambda2 - lambda1)), sin(phi2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-lambda2))
	t_1 = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), t_0)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi1 <= -7.2e-6)
		tmp = t_1;
	elseif (phi1 <= 2.4e-5)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * t_0))), fma(Float64(-Float64(cos(phi2) * phi1)), cos(Float64(lambda2 - lambda1)), sin(phi2)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[(-lambda2)], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -7.2e-6], t$95$1, If[LessEqual[phi1, 2.4e-5], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.19999999999999967e-6 or 2.4000000000000001e-5 < phi1

   1. Initial program 76.5%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 79.3

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

   4. Applied egg-rr 79.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. sin-lowering-sin.f64 N/A

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

      2. neg-lowering-neg.f64 77.5

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

   7. Simplified 77.5%

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

   if -7.19999999999999967e-6 < phi1 < 2.4000000000000001e-5

   1. Initial program 81.9%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. sub-neg N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. cos-neg N/A

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

      15. cos-lowering-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

      19. sin-lowering-sin.f64 99.4

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

   7. Simplified 99.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \left(-\lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \phi_1, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \left(-\lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := -\cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\sin \phi_1 \cdot t\_0, t\_1, t\_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \phi_1, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\sin \phi_1, t\_0 \cdot t\_1, t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (- (cos phi2)))
        (t_2 (* (cos phi1) (sin phi2)))
        (t_3 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.15e-7)
     (atan2 t_3 (fma (* (sin phi1) t_0) t_1 t_2))
     (if (<= phi1 1.52e-5)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (fma (- (* (cos phi2) phi1)) (cos (- lambda2 lambda1)) (sin phi2)))
       (atan2 t_3 (fma (sin phi1) (* t_0 t_1) t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = -cos(phi2);
	double t_2 = cos(phi1) * sin(phi2);
	double t_3 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.15e-7) {
		tmp = atan2(t_3, fma((sin(phi1) * t_0), t_1, t_2));
	} else if (phi1 <= 1.52e-5) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), fma(-(cos(phi2) * phi1), cos((lambda2 - lambda1)), sin(phi2)));
	} else {
		tmp = atan2(t_3, fma(sin(phi1), (t_0 * t_1), t_2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(-cos(phi2))
	t_2 = Float64(cos(phi1) * sin(phi2))
	t_3 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.15e-7)
		tmp = atan(t_3, fma(Float64(sin(phi1) * t_0), t_1, t_2));
	elseif (phi1 <= 1.52e-5)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), fma(Float64(-Float64(cos(phi2) * phi1)), cos(Float64(lambda2 - lambda1)), sin(phi2)));
	else
		tmp = atan(t_3, fma(sin(phi1), Float64(t_0 * t_1), t_2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[Cos[phi2], $MachinePrecision])}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.15e-7], N[ArcTan[t$95$3 / N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.52e-5], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$3 / N[(N[Sin[phi1], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.14999999999999997e-7

   1. Initial program 76.2%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 76.2%

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. sin-lowering-sin.f64 76.2

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

   6. Applied egg-rr 76.2%

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

   if -1.14999999999999997e-7 < phi1 < 1.52e-5

   1. Initial program 81.9%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. sub-neg N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. cos-neg N/A

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

      15. cos-lowering-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

      19. sin-lowering-sin.f64 99.4

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

   7. Simplified 99.4%

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

   if 1.52e-5 < phi1

   1. Initial program 77.0%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 77.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), -\cos \phi_2, \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \phi_1, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := -\cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\sin \phi_1 \cdot t\_0, t\_1, t\_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.02 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\sin \phi_1, t\_0 \cdot t\_1, t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (- (cos phi2)))
        (t_2 (* (cos phi1) (sin phi2)))
        (t_3 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -7.8e-10)
     (atan2 t_3 (fma (* (sin phi1) t_0) t_1 t_2))
     (if (<= phi1 1.02e-104)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       (atan2 t_3 (fma (sin phi1) (* t_0 t_1) t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = -cos(phi2);
	double t_2 = cos(phi1) * sin(phi2);
	double t_3 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -7.8e-10) {
		tmp = atan2(t_3, fma((sin(phi1) * t_0), t_1, t_2));
	} else if (phi1 <= 1.02e-104) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = atan2(t_3, fma(sin(phi1), (t_0 * t_1), t_2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(-cos(phi2))
	t_2 = Float64(cos(phi1) * sin(phi2))
	t_3 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -7.8e-10)
		tmp = atan(t_3, fma(Float64(sin(phi1) * t_0), t_1, t_2));
	elseif (phi1 <= 1.02e-104)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = atan(t_3, fma(sin(phi1), Float64(t_0 * t_1), t_2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[Cos[phi2], $MachinePrecision])}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -7.8e-10], N[ArcTan[t$95$3 / N[(N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.02e-104], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$3 / N[(N[Sin[phi1], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -7.7999999999999999e-10

   1. Initial program 76.2%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 76.2%

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. sin-lowering-sin.f64 76.2

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

   6. Applied egg-rr 76.2%

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

   if -7.7999999999999999e-10 < phi1 < 1.02000000000000001e-104

   1. Initial program 78.5%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. sin-lowering-sin.f64 98.7

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

   7. Simplified 98.7%

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

   if 1.02000000000000001e-104 < phi1

   1. Initial program 82.8%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 82.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), -\cos \phi_2, \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.02 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot t\_0\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\sin \phi_1, t\_0 \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.8e-12)
     (atan2
      t_1
      (fma (cos phi1) (sin phi2) (* (cos phi2) (* (- (sin phi1)) t_0))))
     (if (<= phi1 1.55e-104)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       (atan2
        t_1
        (fma (sin phi1) (* t_0 (- (cos phi2))) (* (cos phi1) (sin phi2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.8e-12) {
		tmp = atan2(t_1, fma(cos(phi1), sin(phi2), (cos(phi2) * (-sin(phi1) * t_0))));
	} else if (phi1 <= 1.55e-104) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = atan2(t_1, fma(sin(phi1), (t_0 * -cos(phi2)), (cos(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.8e-12)
		tmp = atan(t_1, fma(cos(phi1), sin(phi2), Float64(cos(phi2) * Float64(Float64(-sin(phi1)) * t_0))));
	elseif (phi1 <= 1.55e-104)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = atan(t_1, fma(sin(phi1), Float64(t_0 * Float64(-cos(phi2))), Float64(cos(phi1) * sin(phi2))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.8e-12], N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[((-N[Sin[phi1], $MachinePrecision]) * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.55e-104], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi1], $MachinePrecision] * N[(t$95$0 * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.8e-12

   1. Initial program 76.2%

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

   3. Taylor expanded in lambda1 around 0

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. neg-lowering-neg.f64 N/A

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

      18. sin-lowering-sin.f64 76.2

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

   5. Simplified 76.2%

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

   if -1.8e-12 < phi1 < 1.54999999999999988e-104

   1. Initial program 78.5%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. sin-lowering-sin.f64 98.7

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

   7. Simplified 98.7%

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

   if 1.54999999999999988e-104 < phi1

   1. Initial program 82.8%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 82.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot t\_0\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.8e-11)
     (atan2
      t_1
      (fma (cos phi1) (sin phi2) (* (cos phi2) (* (- (sin phi1)) t_0))))
     (if (<= phi1 1.55e-104)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       (atan2
        t_1
        (- (* (cos phi1) (sin phi2)) (* (* (cos phi2) (sin 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 tmp;
	if (phi1 <= -2.8e-11) {
		tmp = atan2(t_1, fma(cos(phi1), sin(phi2), (cos(phi2) * (-sin(phi1) * t_0))));
	} else if (phi1 <= 1.55e-104) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(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)))
	tmp = 0.0
	if (phi1 <= -2.8e-11)
		tmp = atan(t_1, fma(cos(phi1), sin(phi2), Float64(cos(phi2) * Float64(Float64(-sin(phi1)) * t_0))));
	elseif (phi1 <= 1.55e-104)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * 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[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.8e-11], N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[((-N[Sin[phi1], $MachinePrecision]) * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.55e-104], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.8e-11

   1. Initial program 76.2%

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

   3. Taylor expanded in lambda1 around 0

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. neg-lowering-neg.f64 N/A

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

      18. sin-lowering-sin.f64 76.2

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

   5. Simplified 76.2%

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

   if -2.8e-11 < phi1 < 1.54999999999999988e-104

   1. Initial program 78.5%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. sin-lowering-sin.f64 98.7

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

   7. Simplified 98.7%

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

   if 1.54999999999999988e-104 < phi1

   1. Initial program 82.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (* (cos phi2) (sin (- lambda1 lambda2)))
          (fma
           (cos phi1)
           (sin phi2)
           (* (cos phi2) (* (- (sin phi1)) (cos (- lambda1 lambda2))))))))
   (if (<= phi1 -3e-11)
     t_0
     (if (<= phi1 1.55e-104)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((cos(phi2) * sin((lambda1 - lambda2))), fma(cos(phi1), sin(phi2), (cos(phi2) * (-sin(phi1) * cos((lambda1 - lambda2))))));
	double tmp;
	if (phi1 <= -3e-11) {
		tmp = t_0;
	} else if (phi1 <= 1.55e-104) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(cos(phi1), sin(phi2), Float64(cos(phi2) * Float64(Float64(-sin(phi1)) * cos(Float64(lambda1 - lambda2))))))
	tmp = 0.0
	if (phi1 <= -3e-11)
		tmp = t_0;
	elseif (phi1 <= 1.55e-104)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[((-N[Sin[phi1], $MachinePrecision]) * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3e-11], t$95$0, If[LessEqual[phi1, 1.55e-104], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3e-11 or 1.54999999999999988e-104 < phi1

   1. Initial program 79.4%

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

   3. Taylor expanded in lambda1 around 0

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. neg-lowering-neg.f64 N/A

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

      18. sin-lowering-sin.f64 79.4

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

   5. Simplified 79.4%

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

   if -3e-11 < phi1 < 1.54999999999999988e-104

   1. Initial program 78.5%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. sin-lowering-sin.f64 98.7

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

   7. Simplified 98.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2 \cdot \left(-\sin \phi_1\right), t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= lambda1 -3.1e-7)
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
      (sin phi2))
     (if (<= lambda1 1.02e-8)
       (atan2 t_1 (fma (cos lambda2) (* (cos phi2) (- (sin phi1))) t_0))
       (atan2 t_1 (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (lambda1 <= -3.1e-7) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else if (lambda1 <= 1.02e-8) {
		tmp = atan2(t_1, fma(cos(lambda2), (cos(phi2) * -sin(phi1)), t_0));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (lambda1 <= -3.1e-7)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	elseif (lambda1 <= 1.02e-8)
		tmp = atan(t_1, fma(cos(lambda2), Float64(cos(phi2) * Float64(-sin(phi1))), t_0));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.1e-7], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 1.02e-8], N[ArcTan[t$95$1 / N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.1e-7

   1. Initial program 59.0%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. neg-lowering-neg.f64 80.9

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

   4. Applied egg-rr 80.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. sin-lowering-sin.f64 60.6

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

   7. Simplified 60.6%

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

   if -3.1e-7 < lambda1 < 1.02000000000000003e-8

   1. Initial program 99.4%

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

      1. atan2-lowering-atan2.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. sin-neg N/A

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

      5. remove-double-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. cos-neg N/A

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

      9. cos-lowering-cos.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. mul-1-neg N/A

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

      13. neg-lowering-neg.f64 N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. cos-lowering-cos.f64 N/A

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

      17. sin-lowering-sin.f64 99.4

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

   10. Simplified 99.4%

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

   if 1.02000000000000003e-8 < lambda1

   1. Initial program 62.1%

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

   3. Taylor expanded in lambda1 around inf

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

   5. Simplified 61.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \lambda_2, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2 - \cos \lambda_2 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2 - \cos \lambda_1 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (* (cos phi1) (sin phi2))))
   (if (<= lambda1 -3.1e-7)
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
      (sin phi2))
     (if (<= lambda1 1.02e-8)
       (atan2 t_1 (- t_2 (* (cos lambda2) t_0)))
       (atan2 t_1 (- t_2 (* (cos lambda1) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -3.1e-7) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else if (lambda1 <= 1.02e-8) {
		tmp = atan2(t_1, (t_2 - (cos(lambda2) * t_0)));
	} else {
		tmp = atan2(t_1, (t_2 - (cos(lambda1) * t_0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -3.1e-7)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	elseif (lambda1 <= 1.02e-8)
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda2) * t_0)));
	else
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda1) * t_0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $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[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.1e-7], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 1.02e-8], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.1e-7

   1. Initial program 59.0%

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

      1. sin-diff N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sin-neg N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. neg-lowering-neg.f64 80.9

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(-\lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied egg-rr 80.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 60.6

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 60.6%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -3.1e-7 < lambda1 < 1.02000000000000003e-8

   1. Initial program 99.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

      2. cos-lowering-cos.f64 99.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

   5. Simplified 99.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_2}} \]

   if 1.02000000000000003e-8 < lambda1

   1. Initial program 62.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]
   4. Step-by-step derivation

   5. Simplified 61.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -5800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{+41}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (*
           (cos phi2)
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (cos lambda1) (sin (- lambda2)))))
          (sin phi2))))
   (if (<= lambda2 -5800000.0)
     t_0
     (if (<= lambda2 1.95e+41)
       (atan2
        (* (cos phi2) (sin (- lambda1 lambda2)))
        (-
         (* (cos phi1) (sin phi2))
         (* (cos lambda1) (* (cos phi2) (sin phi1)))))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	double tmp;
	if (lambda2 <= -5800000.0) {
		tmp = t_0;
	} else if (lambda2 <= 1.95e+41) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2))
	tmp = 0.0
	if (lambda2 <= -5800000.0)
		tmp = t_0;
	elseif (lambda2 <= 1.95e+41)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -5800000.0], t$95$0, If[LessEqual[lambda2, 1.95e+41], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -5.8e6 or 1.9499999999999998e41 < lambda2

   1. Initial program 61.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. neg-lowering-neg.f64 80.5

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(-\lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied egg-rr 80.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 58.9

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 58.9%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -5.8e6 < lambda2 < 1.9499999999999998e41

   1. Initial program 97.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]
   4. Step-by-step derivation

   5. Simplified 93.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\lambda_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5800000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{+41}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot t\_0}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(-\sin \phi_1\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.9e-11)
     (atan2 t_1 (- (* (cos phi1) (sin phi2)) (* (sin phi1) t_0)))
     (if (<= phi1 1.55e-104)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       (atan2 t_1 (fma (sin phi2) (cos phi1) (* (- (sin 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 tmp;
	if (phi1 <= -1.9e-11) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)));
	} else if (phi1 <= 1.55e-104) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = atan2(t_1, fma(sin(phi2), cos(phi1), (-sin(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)))
	tmp = 0.0
	if (phi1 <= -1.9e-11)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * t_0)));
	elseif (phi1 <= 1.55e-104)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = atan(t_1, fma(sin(phi2), cos(phi1), Float64(Float64(-sin(phi1)) * 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[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.9e-11], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.55e-104], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[((-N[Sin[phi1], $MachinePrecision]) * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.8999999999999999e-11

   1. Initial program 76.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \sin \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \sin \phi_1} \]

      4. sin-lowering-sin.f64 51.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

   5. Simplified 51.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   if -1.8999999999999999e-11 < phi1 < 1.54999999999999988e-104

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. neg-lowering-neg.f64 99.8

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(-\lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 98.7

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 98.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if 1.54999999999999988e-104 < phi1

   1. Initial program 82.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\cos \phi_2\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\cos \phi_2\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\cos \phi_2\right)\right)\right)\right)}} \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\color{blue}{\sin \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\cos \phi_2\right)\right)\right)\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\cos \phi_2\right)\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\cos \phi_2\right)\right)\right) \cdot \sin \phi_1}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\mathsf{neg}\left(\cos \phi_2\right)\right) \cdot \sin \phi_1\right)}\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\right)} \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \cos \phi_2}\right)\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\right)} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \left(\mathsf{neg}\left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \phi_2 \cdot \sin \phi_1}\right)\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \left(\mathsf{neg}\left(\sin \phi_1\right)\right)\right)}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \left(\mathsf{neg}\left(\sin \phi_1\right)\right)\right)}\right)} \]

      16. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\cos \phi_2} \cdot \left(\mathsf{neg}\left(\sin \phi_1\right)\right)\right)\right)} \]

      17. neg-lowering-neg.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \phi_1\right)\right)}\right)\right)} \]

      18. sin-lowering-sin.f64 82.7

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \left(-\color{blue}{\sin \phi_1}\right)\right)\right)} \]

   6. Applied egg-rr 82.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)\right)}} \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \mathsf{neg}\left(\color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \mathsf{neg}\left(\sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)} \]

      7. --lowering--.f64 60.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, -\sin \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)} \]

   9. Simplified 60.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \color{blue}{-\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \left(-\sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (* (cos phi2) (sin (- lambda1 lambda2)))
          (-
           (* (cos phi1) (sin phi2))
           (* (sin phi1) (cos (- lambda1 lambda2)))))))
   (if (<= phi1 -2.15e-10)
     t_0
     (if (<= phi1 1.55e-104)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi1 <= -2.15e-10) {
		tmp = t_0;
	} else if (phi1 <= 1.55e-104) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (phi1 <= -2.15e-10)
		tmp = t_0;
	elseif (phi1 <= 1.55e-104)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -2.15e-10], t$95$0, If[LessEqual[phi1, 1.55e-104], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.15000000000000007e-10 or 1.54999999999999988e-104 < phi1

   1. Initial program 79.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \sin \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \sin \phi_1} \]

      4. sin-lowering-sin.f64 55.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sin \phi_1}} \]

   5. Simplified 55.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   if -2.15000000000000007e-10 < phi1 < 1.54999999999999988e-104

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. neg-lowering-neg.f64 99.8

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(-\lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 98.7

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 98.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (atan2
          (* (cos phi2) (sin (- lambda1 lambda2)))
          (* (- (sin phi1)) (cos (- lambda2 lambda1))))))
   (if (<= phi1 -1.15e-6)
     t_0
     (if (<= phi1 2.8e-71)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2)))))
        (sin phi2))
       t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2((cos(phi2) * sin((lambda1 - lambda2))), (-sin(phi1) * cos((lambda2 - lambda1))));
	double tmp;
	if (phi1 <= -1.15e-6) {
		tmp = t_0;
	} else if (phi1 <= 2.8e-71) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2)))), sin(phi2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(-sin(phi1)) * cos(Float64(lambda2 - lambda1))))
	tmp = 0.0
	if (phi1 <= -1.15e-6)
		tmp = t_0;
	elseif (phi1 <= 2.8e-71)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * sin(Float64(-lambda2))))), sin(phi2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[((-N[Sin[phi1], $MachinePrecision]) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.15e-6], t$95$0, If[LessEqual[phi1, 2.8e-71], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.15e-6 or 2.8e-71 < phi1

   1. Initial program 78.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)} \]

      4. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      5. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      8. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \sin \phi_1\right)} \]

      9. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \sin \phi_1\right)} \]

      12. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      13. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      14. sin-lowering-sin.f64 50.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}} \]

   7. Simplified 50.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]

   if -1.15e-6 < phi1 < 2.8e-71

   1. Initial program 79.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \cos \lambda_2, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \color{blue}{\cos \lambda_2}, \mathsf{neg}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \left(\mathsf{neg}\left(\sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      7. sin-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \color{blue}{\cos \lambda_1} \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      11. neg-lowering-neg.f64 99.8

          \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \color{blue}{\left(-\lambda_2\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 98.8

         \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 98.8%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t\_1\\ \mathbf{if}\;\phi_1 \leq -0.0015:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\left(-\sin \phi_1\right) \cdot t\_0}\\ \mathbf{elif}\;\phi_1 \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(-\cos \phi_2 \cdot \phi_1, t\_0, \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* (cos phi2) t_1)))
   (if (<= phi1 -0.0015)
     (atan2 t_2 (* (- (sin phi1)) t_0))
     (if (<= phi1 1.65e-9)
       (atan2 t_2 (fma (- (* (cos phi2) phi1)) t_0 (sin phi2)))
       (atan2
        t_1
        (-
         (* (cos phi1) (sin phi2))
         (* (sin phi1) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = cos(phi2) * t_1;
	double tmp;
	if (phi1 <= -0.0015) {
		tmp = atan2(t_2, (-sin(phi1) * t_0));
	} else if (phi1 <= 1.65e-9) {
		tmp = atan2(t_2, fma(-(cos(phi2) * phi1), t_0, sin(phi2)));
	} else {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * t_1)
	tmp = 0.0
	if (phi1 <= -0.0015)
		tmp = atan(t_2, Float64(Float64(-sin(phi1)) * t_0));
	elseif (phi1 <= 1.65e-9)
		tmp = atan(t_2, fma(Float64(-Float64(cos(phi2) * phi1)), t_0, sin(phi2)));
	else
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[phi1, -0.0015], N[ArcTan[t$95$2 / N[((-N[Sin[phi1], $MachinePrecision]) * t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.65e-9], N[ArcTan[t$95$2 / N[((-N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]) * t$95$0 + N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0015

   1. Initial program 76.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)} \]

      4. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      5. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      8. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \sin \phi_1\right)} \]

      9. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \sin \phi_1\right)} \]

      12. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      13. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      14. sin-lowering-sin.f64 49.2

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}} \]

   7. Simplified 49.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]

   if -0.0015 < phi1 < 1.65000000000000009e-9

   1. Initial program 82.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2 + -1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \sin \phi_2}} \]

      2. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(-1 \cdot \phi_1\right) \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + \sin \phi_2} \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\left(-1 \cdot \phi_1\right) \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \sin \phi_2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\left(-1 \cdot \phi_1\right) \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\color{blue}{\left(-1 \cdot \phi_1\right) \cdot \cos \phi_2}, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_2\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\phi_1\right)\right)} \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_2\right)} \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\phi_1\right)\right)} \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_2\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \color{blue}{\cos \phi_2}, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_2\right)} \]

      9. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}, \sin \phi_2\right)} \]

      10. remove-double-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \sin \phi_2\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \sin \phi_2\right)} \]

      12. distribute-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}, \sin \phi_2\right)} \]

      13. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right), \sin \phi_2\right)} \]

      14. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}, \sin \phi_2\right)} \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}, \sin \phi_2\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), \sin \phi_2\right)} \]

      17. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \sin \phi_2\right)} \]

      18. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\phi_1\right)\right) \cdot \cos \phi_2, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \sin \phi_2\right)} \]

      19. sin-lowering-sin.f64 82.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(-\phi_1\right) \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \color{blue}{\sin \phi_2}\right)} \]

   7. Simplified 82.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\left(-\phi_1\right) \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2\right)}} \]

   if 1.65000000000000009e-9 < phi1

   1. Initial program 76.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 41.6

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 41.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 41.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Simplified 41.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0015:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(-\cos \phi_2 \cdot \phi_1, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.0095:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* (cos phi2) t_0) (sin phi2))))
   (if (<= phi2 -0.0095)
     t_1
     (if (<= phi2 0.00095)
       (atan2
        t_0
        (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda1 lambda2)))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((cos(phi2) * t_0), sin(phi2));
	double tmp;
	if (phi2 <= -0.0095) {
		tmp = t_1;
	} else if (phi2 <= 0.00095) {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = atan2((cos(phi2) * t_0), sin(phi2))
    if (phi2 <= (-0.0095d0)) then
        tmp = t_1
    else if (phi2 <= 0.00095d0) then
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.atan2((Math.cos(phi2) * t_0), Math.sin(phi2));
	double tmp;
	if (phi2 <= -0.0095) {
		tmp = t_1;
	} else if (phi2 <= 0.00095) {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.atan2((math.cos(phi2) * t_0), math.sin(phi2))
	tmp = 0
	if phi2 <= -0.0095:
		tmp = t_1
	elif phi2 <= 0.00095:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(cos(phi2) * t_0), sin(phi2))
	tmp = 0.0
	if (phi2 <= -0.0095)
		tmp = t_1;
	elseif (phi2 <= 0.00095)
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = atan2((cos(phi2) * t_0), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -0.0095)
		tmp = t_1;
	elseif (phi2 <= 0.00095)
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.0095], t$95$1, If[LessEqual[phi2, 0.00095], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.00949999999999999976 or 9.49999999999999998e-4 < phi2

   1. Initial program 72.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 42.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 42.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -0.00949999999999999976 < phi2 < 9.49999999999999998e-4

   1. Initial program 87.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 87.0

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 87.0%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 87.0

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   8. Simplified 87.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0095:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 64.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 0.0004:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* (cos phi2) t_0) (sin phi2))))
   (if (<= phi2 -0.003)
     t_1
     (if (<= phi2 0.0004)
       (atan2
        t_0
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1)))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((cos(phi2) * t_0), sin(phi2));
	double tmp;
	if (phi2 <= -0.003) {
		tmp = t_1;
	} else if (phi2 <= 0.0004) {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = atan2((cos(phi2) * t_0), sin(phi2))
    if (phi2 <= (-0.003d0)) then
        tmp = t_1
    else if (phi2 <= 0.0004d0) then
        tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.atan2((Math.cos(phi2) * t_0), Math.sin(phi2));
	double tmp;
	if (phi2 <= -0.003) {
		tmp = t_1;
	} else if (phi2 <= 0.0004) {
		tmp = Math.atan2(t_0, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.atan2((math.cos(phi2) * t_0), math.sin(phi2))
	tmp = 0
	if phi2 <= -0.003:
		tmp = t_1
	elif phi2 <= 0.0004:
		tmp = math.atan2(t_0, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(cos(phi2) * t_0), sin(phi2))
	tmp = 0.0
	if (phi2 <= -0.003)
		tmp = t_1;
	elseif (phi2 <= 0.0004)
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = atan2((cos(phi2) * t_0), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -0.003)
		tmp = t_1;
	elseif (phi2 <= 0.0004)
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.003], t$95$1, If[LessEqual[phi2, 0.0004], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0030000000000000001 or 4.00000000000000019e-4 < phi2

   1. Initial program 72.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 42.1

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 42.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -0.0030000000000000001 < phi2 < 4.00000000000000019e-4

   1. Initial program 87.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 87.0

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 87.0%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \color{blue}{\cos \phi_1} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      5. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \sin \phi_1} \]

      6. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1} \]

      7. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1} \]

      8. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \sin \phi_1} \]

      9. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \sin \phi_1} \]

      10. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1} \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \sin \phi_1} \]

      13. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1} \]

      14. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1} \]

      15. sin-lowering-sin.f64 87.0

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}} \]

   8. Simplified 87.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 0.0004:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 62.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_1 (atan2 t_0 (* (- (sin phi1)) (cos (- lambda2 lambda1))))))
   (if (<= phi1 -4.5e-8)
     t_1
     (if (<= phi1 2.8e-71) (atan2 t_0 (sin phi2)) t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double t_1 = atan2(t_0, (-sin(phi1) * cos((lambda2 - lambda1))));
	double tmp;
	if (phi1 <= -4.5e-8) {
		tmp = t_1;
	} else if (phi1 <= 2.8e-71) {
		tmp = atan2(t_0, sin(phi2));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    t_1 = atan2(t_0, (-sin(phi1) * cos((lambda2 - lambda1))))
    if (phi1 <= (-4.5d-8)) then
        tmp = t_1
    else if (phi1 <= 2.8d-71) then
        tmp = atan2(t_0, sin(phi2))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_1 = Math.atan2(t_0, (-Math.sin(phi1) * Math.cos((lambda2 - lambda1))));
	double tmp;
	if (phi1 <= -4.5e-8) {
		tmp = t_1;
	} else if (phi1 <= 2.8e-71) {
		tmp = Math.atan2(t_0, Math.sin(phi2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	t_1 = math.atan2(t_0, (-math.sin(phi1) * math.cos((lambda2 - lambda1))))
	tmp = 0
	if phi1 <= -4.5e-8:
		tmp = t_1
	elif phi1 <= 2.8e-71:
		tmp = math.atan2(t_0, math.sin(phi2))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_1 = atan(t_0, Float64(Float64(-sin(phi1)) * cos(Float64(lambda2 - lambda1))))
	tmp = 0.0
	if (phi1 <= -4.5e-8)
		tmp = t_1;
	elseif (phi1 <= 2.8e-71)
		tmp = atan(t_0, sin(phi2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	t_1 = atan2(t_0, (-sin(phi1) * cos((lambda2 - lambda1))));
	tmp = 0.0;
	if (phi1 <= -4.5e-8)
		tmp = t_1;
	elseif (phi1 <= 2.8e-71)
		tmp = atan2(t_0, sin(phi2));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[t$95$0 / N[((-N[Sin[phi1], $MachinePrecision]) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -4.5e-8], t$95$1, If[LessEqual[phi1, 2.8e-71], N[ArcTan[t$95$0 / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.49999999999999993e-8 or 2.8e-71 < phi1

   1. Initial program 78.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)} \]

      4. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      5. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      8. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \sin \phi_1\right)} \]

      9. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \sin \phi_1\right)} \]

      12. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      13. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      14. sin-lowering-sin.f64 50.5

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}} \]

   7. Simplified 50.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]

   if -4.49999999999999993e-8 < phi1 < 2.8e-71

   1. Initial program 79.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 78.8

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 78.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* (cos phi2) t_0) (sin phi2))))
   (if (<= phi2 -2.8e-14)
     t_1
     (if (<= phi2 5.3e-8)
       (atan2 t_0 (* (- (sin phi1)) (cos (- lambda2 lambda1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((cos(phi2) * t_0), sin(phi2));
	double tmp;
	if (phi2 <= -2.8e-14) {
		tmp = t_1;
	} else if (phi2 <= 5.3e-8) {
		tmp = atan2(t_0, (-sin(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = atan2((cos(phi2) * t_0), sin(phi2))
    if (phi2 <= (-2.8d-14)) then
        tmp = t_1
    else if (phi2 <= 5.3d-8) then
        tmp = atan2(t_0, (-sin(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.atan2((Math.cos(phi2) * t_0), Math.sin(phi2));
	double tmp;
	if (phi2 <= -2.8e-14) {
		tmp = t_1;
	} else if (phi2 <= 5.3e-8) {
		tmp = Math.atan2(t_0, (-Math.sin(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.atan2((math.cos(phi2) * t_0), math.sin(phi2))
	tmp = 0
	if phi2 <= -2.8e-14:
		tmp = t_1
	elif phi2 <= 5.3e-8:
		tmp = math.atan2(t_0, (-math.sin(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(cos(phi2) * t_0), sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.8e-14)
		tmp = t_1;
	elseif (phi2 <= 5.3e-8)
		tmp = atan(t_0, Float64(Float64(-sin(phi1)) * cos(Float64(lambda2 - lambda1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = atan2((cos(phi2) * t_0), sin(phi2));
	tmp = 0.0;
	if (phi2 <= -2.8e-14)
		tmp = t_1;
	elseif (phi2 <= 5.3e-8)
		tmp = atan2(t_0, (-sin(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -2.8e-14], t$95$1, If[LessEqual[phi2, 5.3e-8], N[ArcTan[t$95$0 / N[((-N[Sin[phi1], $MachinePrecision]) * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.8000000000000001e-14 or 5.2999999999999998e-8 < phi2

   1. Initial program 72.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. atan2-lowering-atan2.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      6. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

      8. associate-*l* N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   4. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   6. Step-by-step derivation

      1. sin-lowering-sin.f64 42.5

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   7. Simplified 42.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -2.8000000000000001e-14 < phi2 < 5.2999999999999998e-8

   1. Initial program 86.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 86.9

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 86.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{neg}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)} \]

      4. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      5. remove-double-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \sin \phi_1\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \sin \phi_1\right)} \]

      8. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \sin \phi_1\right)} \]

      9. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \sin \phi_1\right)} \]

      12. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      13. --lowering--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{neg}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \sin \phi_1\right)} \]

      14. sin-lowering-sin.f64 84.0

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\sin \phi_1}} \]

   8. Simplified 84.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{-\cos \left(\lambda_2 - \lambda_1\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\left(-\sin \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 48.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * sin((lambda1 - lambda2))), sin(phi2));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), sin(phi2))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), Math.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. atan2-lowering-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. --lowering--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. sub-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

   7. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\mathsf{neg}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \sin \phi_2}} \]

   8. associate-*l* N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\left(\mathsf{neg}\left(\color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) + \cos \phi_1 \cdot \sin \phi_2} \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(\mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \sin \phi_2} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \mathsf{neg}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

4. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2\right), \cos \phi_1 \cdot \sin \phi_2\right)}} \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
6. Step-by-step derivation

   1. sin-lowering-sin.f64 47.7

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

7. Simplified 47.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

8. Final simplification 47.7%

   \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]
9. Add Preprocessing

Alternative 28: 31.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\left(\lambda_2 + \lambda_1\right) \cdot \frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1}\right)}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (sin (* (+ lambda2 lambda1) (/ (- lambda1 lambda2) (+ lambda2 lambda1))))
  (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin(((lambda2 + lambda1) * ((lambda1 - lambda2) / (lambda2 + lambda1)))), sin(phi2));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2(sin(((lambda2 + lambda1) * ((lambda1 - lambda2) / (lambda2 + lambda1)))), sin(phi2))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin(((lambda2 + lambda1) * ((lambda1 - lambda2) / (lambda2 + lambda1)))), Math.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin(((lambda2 + lambda1) * ((lambda1 - lambda2) / (lambda2 + lambda1)))), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(Float64(lambda2 + lambda1) * Float64(Float64(lambda1 - lambda2) / Float64(lambda2 + lambda1)))), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin(((lambda2 + lambda1) * ((lambda1 - lambda2) / (lambda2 + lambda1)))), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(N[(lambda2 + lambda1), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] / N[(lambda2 + lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. --lowering--.f64 49.5

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Simplified 49.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
7. Step-by-step derivation

   1. sin-lowering-sin.f64 32.9

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

8. Simplified 32.9%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
9. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)}}{\sin \phi_2} \]

   2. difference-of-squares N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}{\lambda_1 + \lambda_2}\right)}{\sin \phi_2} \]

   3. associate-/l* N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\left(\lambda_1 + \lambda_2\right) \cdot \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\right)}}{\sin \phi_2} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2} \cdot \left(\lambda_1 + \lambda_2\right)\right)}}{\sin \phi_2} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2} \cdot \left(\lambda_1 + \lambda_2\right)\right)}}{\sin \phi_2} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\color{blue}{\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}} \cdot \left(\lambda_1 + \lambda_2\right)\right)}{\sin \phi_2} \]

   7. --lowering--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{\lambda_1 + \lambda_2} \cdot \left(\lambda_1 + \lambda_2\right)\right)}{\sin \phi_2} \]

   8. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\color{blue}{\lambda_2 + \lambda_1}} \cdot \left(\lambda_1 + \lambda_2\right)\right)}{\sin \phi_2} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\color{blue}{\lambda_2 + \lambda_1}} \cdot \left(\lambda_1 + \lambda_2\right)\right)}{\sin \phi_2} \]

   10. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \color{blue}{\left(\lambda_2 + \lambda_1\right)}\right)}{\sin \phi_2} \]

   11. +-lowering-+.f64 33.0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \color{blue}{\left(\lambda_2 + \lambda_1\right)}\right)}{\sin \phi_2} \]

10. Applied egg-rr 33.0%

    \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1} \cdot \left(\lambda_2 + \lambda_1\right)\right)}}{\sin \phi_2} \]

11. Final simplification 33.0%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\left(\lambda_2 + \lambda_1\right) \cdot \frac{\lambda_1 - \lambda_2}{\lambda_2 + \lambda_1}\right)}{\sin \phi_2} \]
12. Add Preprocessing

Alternative 29: 31.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), sin(phi2));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2(sin((lambda1 - lambda2)), sin(phi2))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin((lambda1 - lambda2)), Math.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. --lowering--.f64 49.5

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Simplified 49.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
7. Step-by-step derivation

   1. sin-lowering-sin.f64 32.9

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

8. Simplified 32.9%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
9. Add Preprocessing

Alternative 30: 31.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -2.8e-15)
   (atan2 (sin lambda1) phi2)
   (atan2
    (sin (- lambda1 lambda2))
    (fma
     phi2
     (*
      (* phi2 phi2)
      (fma
       (* phi2 phi2)
       (fma (* phi2 phi2) -0.0001984126984126984 0.008333333333333333)
       -0.16666666666666666))
     phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.8e-15) {
		tmp = atan2(sin(lambda1), phi2);
	} else {
		tmp = atan2(sin((lambda1 - lambda2)), fma(phi2, ((phi2 * phi2) * fma((phi2 * phi2), fma((phi2 * phi2), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -2.8e-15)
		tmp = atan(sin(lambda1), phi2);
	else
		tmp = atan(sin(Float64(lambda1 - lambda2)), fma(phi2, Float64(Float64(phi2 * phi2) * fma(Float64(phi2 * phi2), fma(Float64(phi2 * phi2), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -2.8e-15], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(phi2 * N[(N[(phi2 * phi2), $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + phi2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.80000000000000014e-15

   1. Initial program 67.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 14.1

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 14.1%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 12.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   8. Simplified 12.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
   10. Step-by-step derivation

   11. Simplified 12.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
   13. Step-by-step derivation

       1. sin-lowering-sin.f64 13.3

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

   14. Simplified 13.3%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

   if -2.80000000000000014e-15 < phi2

   1. Initial program 83.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 61.6

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 61.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 39.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   8. Simplified 39.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \color{blue}{\left({\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \left({\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) + \phi_2 \cdot 1}} \]

       3. *-rgt-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \left({\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{\phi_2}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\phi_2, {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right), \phi_2\right)}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{{\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)}, \phi_2\right)} \]

       6. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{\left(\phi_2 \cdot \phi_2\right)} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right), \phi_2\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{\left(\phi_2 \cdot \phi_2\right)} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right), \phi_2\right)} \]

       8. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \color{blue}{\left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, \phi_2\right)} \]

       9. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), \phi_2\right)} \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left({\phi_2}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}, \frac{-1}{6}\right)}, \phi_2\right)} \]

       11. unpow2 N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\phi_2 \cdot \phi_2}, \frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}, \frac{-1}{6}\right), \phi_2\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\phi_2 \cdot \phi_2}, \frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}, \frac{-1}{6}\right), \phi_2\right)} \]

       13. +-commutative N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{5040} \cdot {\phi_2}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), \phi_2\right)} \]

       14. *-commutative N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{{\phi_2}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), \phi_2\right)} \]

       15. accelerator-lowering-fma.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), \phi_2\right)} \]

       16. unpow2 N/A

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \mathsf{fma}\left(\color{blue}{\phi_2 \cdot \phi_2}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), \phi_2\right)} \]

       17. *-lowering-*.f64 40.6

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \mathsf{fma}\left(\color{blue}{\phi_2 \cdot \phi_2}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \phi_2\right)} \]

   11. Simplified 40.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), \phi_2\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 31.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot -0.16666666666666666, \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -2.8e-15)
   (atan2 (sin lambda1) phi2)
   (atan2
    (sin (- lambda1 lambda2))
    (fma phi2 (* (* phi2 phi2) -0.16666666666666666) phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.8e-15) {
		tmp = atan2(sin(lambda1), phi2);
	} else {
		tmp = atan2(sin((lambda1 - lambda2)), fma(phi2, ((phi2 * phi2) * -0.16666666666666666), phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -2.8e-15)
		tmp = atan(sin(lambda1), phi2);
	else
		tmp = atan(sin(Float64(lambda1 - lambda2)), fma(phi2, Float64(Float64(phi2 * phi2) * -0.16666666666666666), phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -2.8e-15], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(phi2 * N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + phi2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.80000000000000014e-15

   1. Initial program 67.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 14.1

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 14.1%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 12.4

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   8. Simplified 12.4%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
   10. Step-by-step derivation

   11. Simplified 12.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
   13. Step-by-step derivation

       1. sin-lowering-sin.f64 13.3

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

   14. Simplified 13.3%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

   if -2.80000000000000014e-15 < phi2

   1. Initial program 83.1%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 61.6

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 61.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 39.9

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   8. Simplified 39.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2 \cdot 1}} \]

       3. *-rgt-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \color{blue}{\phi_2}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot {\phi_2}^{2}, \phi_2\right)}} \]

       5. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{{\phi_2}^{2} \cdot \frac{-1}{6}}, \phi_2\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{{\phi_2}^{2} \cdot \frac{-1}{6}}, \phi_2\right)} \]

       7. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{\left(\phi_2 \cdot \phi_2\right)} \cdot \frac{-1}{6}, \phi_2\right)} \]

       8. *-lowering-*.f64 40.6

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\phi_2, \color{blue}{\left(\phi_2 \cdot \phi_2\right)} \cdot -0.16666666666666666, \phi_2\right)} \]

   11. Simplified 40.6%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot -0.16666666666666666, \phi_2\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 27.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1}{\phi_2}\\ \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 4.1 \cdot 10^{-66}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (atan2 (sin lambda1) phi2)))
   (if (<= lambda1 -2.3e-78)
     t_0
     (if (<= lambda1 4.1e-66) (atan2 (sin (- lambda2)) phi2) t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = atan2(sin(lambda1), phi2);
	double tmp;
	if (lambda1 <= -2.3e-78) {
		tmp = t_0;
	} else if (lambda1 <= 4.1e-66) {
		tmp = atan2(sin(-lambda2), phi2);
	} else {
		tmp = 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 = atan2(sin(lambda1), phi2)
    if (lambda1 <= (-2.3d-78)) then
        tmp = t_0
    else if (lambda1 <= 4.1d-66) then
        tmp = atan2(sin(-lambda2), phi2)
    else
        tmp = 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.atan2(Math.sin(lambda1), phi2);
	double tmp;
	if (lambda1 <= -2.3e-78) {
		tmp = t_0;
	} else if (lambda1 <= 4.1e-66) {
		tmp = Math.atan2(Math.sin(-lambda2), phi2);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.atan2(math.sin(lambda1), phi2)
	tmp = 0
	if lambda1 <= -2.3e-78:
		tmp = t_0
	elif lambda1 <= 4.1e-66:
		tmp = math.atan2(math.sin(-lambda2), phi2)
	else:
		tmp = t_0
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = atan(sin(lambda1), phi2)
	tmp = 0.0
	if (lambda1 <= -2.3e-78)
		tmp = t_0;
	elseif (lambda1 <= 4.1e-66)
		tmp = atan(sin(Float64(-lambda2)), phi2);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = atan2(sin(lambda1), phi2);
	tmp = 0.0;
	if (lambda1 <= -2.3e-78)
		tmp = t_0;
	elseif (lambda1 <= 4.1e-66)
		tmp = atan2(sin(-lambda2), phi2);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[ArcTan[N[Sin[lambda1], $MachinePrecision] / phi2], $MachinePrecision]}, If[LessEqual[lambda1, -2.3e-78], t$95$0, If[LessEqual[lambda1, 4.1e-66], N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / phi2], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.3000000000000002e-78 or 4.09999999999999998e-66 < lambda1

   1. Initial program 66.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 43.0

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 43.0%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 31.2

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   8. Simplified 31.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
   10. Step-by-step derivation

   11. Simplified 29.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
   13. Step-by-step derivation

       1. sin-lowering-sin.f64 28.7

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

   14. Simplified 28.7%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

   if -2.3000000000000002e-78 < lambda1 < 4.09999999999999998e-66

   1. Initial program 99.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. --lowering--.f64 60.1

         \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   5. Simplified 60.1%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 35.7

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   8. Simplified 35.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
   10. Step-by-step derivation

   11. Simplified 29.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\phi_2} \]
   13. Step-by-step derivation

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\phi_2} \]

       2. neg-lowering-neg.f64 27.2

          \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(-\lambda_2\right)}}{\phi_2} \]

   14. Simplified 27.2%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(-\lambda_2\right)}}{\phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 29.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) phi2))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), phi2);
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2(sin((lambda1 - lambda2)), phi2)
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin((lambda1 - lambda2)), phi2);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), phi2)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), phi2)
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), phi2);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. --lowering--.f64 49.5

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Simplified 49.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
7. Step-by-step derivation

   1. sin-lowering-sin.f64 32.9

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

8. Simplified 32.9%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
10. Step-by-step derivation

11. Simplified 29.3%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
12. Add Preprocessing

Alternative 34: 22.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \lambda_1}{\phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin lambda1) phi2))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin(lambda1), phi2);
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2(sin(lambda1), phi2)
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin(lambda1), phi2);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin(lambda1), phi2)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(lambda1), phi2)
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin(lambda1), phi2);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[lambda1], $MachinePrecision] / phi2], $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. --lowering--.f64 49.5

      \[\leadsto \tan^{-1}_* \frac{\sin \color{blue}{\left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

5. Simplified 49.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
7. Step-by-step derivation

   1. sin-lowering-sin.f64 32.9

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

8. Simplified 32.9%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
10. Step-by-step derivation

11. Simplified 29.3%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]

12. Taylor expanded in lambda2 around 0

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
13. Step-by-step derivation

    1. sin-lowering-sin.f64 22.5

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]

14. Simplified 22.5%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))