Bearing on a great circle

Percentage Accurate: 79.4% → 99.7%

Time: 30.9s

Alternatives: 29

Speedup: 1.0×

• Could not converge on a ground truth

  1. lambda1 = 1.1730015617630145e+294
  2. lambda2 = 2.9175648403910267e+294
  3. phi1 = 4.923177370884608e+27
  4. phi2 = 1.8294188081992143e-70

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: 79.4% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * fma(sin(lambda2), Float64(-cos(lambda1)), Float64(sin(lambda1) * cos(lambda2)))), Float64(Float64(sin(phi2) * cos(phi1)) - Float64(Float64(sin(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1)))) * cos(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[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 91.7

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

4. Applied rewrites 91.7%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 89.4

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

4. Applied rewrites 89.4%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 99.7

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

9. Applied rewrites 99.7%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. *-commutative N/A

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

    7. lift-*.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Reproduce

herbie shell --seed 2024234 
(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))))))