Bearing on a great circle

Average Error: 12.6 → 0.2

Time: 40.7s

Precision: binary64

Cost: 103744

Math

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

↓

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

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))))))

↓

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

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)))));
}

↓

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

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

↓

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

↓

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[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 12.6

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

2. Simplified 12.6

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   Proof
   (atan2.f64 (*.f64 (sin.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (sin.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 19.0

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

4. Simplified 19.0

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left|\cos \left(\lambda_1 - \lambda_2\right)\right|}\right)} \]
   Proof
   (atan2.f64 (*.f64 (sin.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi2) (fabs.f64 (cos.f64 (-.f64 lambda1 lambda2))))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (sin.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi2) (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (cos.f64 (-.f64 lambda1 lambda2)) (cos.f64 (-.f64 lambda1 lambda2))))))))): 3 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (sin.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi2) (sqrt.f64 (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (-.f64 lambda1 lambda2)) 2))))))): 3 points increase in error, 0 points decrease in error

5. Applied egg-rr 12.8

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

6. Applied egg-rr 0.2

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

7. Simplified 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
   Proof
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi2) (sin.f64 phi1)) (+.f64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 phi1) (cos.f64 phi2))) (+.f64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi2)) (*.f64 (cos.f64 lambda1) (cos.f64 lambda2))) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))))): 0 points increase in error, 0 points decrease in error

8. Taylor expanded in phi2 around inf 0.2

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

9. Simplified 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)}} \]
   Proof
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (fma.f64 (sin.f64 lambda2) (sin.f64 lambda1) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))) (cos.f64 phi2))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (fma.f64 (sin.f64 lambda2) (sin.f64 lambda1) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)))) (cos.f64 phi2))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)))) (cos.f64 phi2))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (*.f64 (sin.f64 phi1) (*.f64 (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))) (cos.f64 phi2))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (sin.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))) (cos.f64 phi2))))): 0 points increase in error, 0 points decrease in error
   (atan2.f64 (*.f64 (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))) (cos.f64 phi2)) (-.f64 (*.f64 (cos.f64 phi1) (sin.f64 phi2)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))): 0 points increase in error, 0 points decrease in error

10. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	97472

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

Alternative 2
Error	0.2
Cost	91136

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

Alternative 3
Error	3.9
Cost	84996

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_3 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_4 := t_0 - t_3\\ \mathbf{if}\;\phi_2 \leq -3.45 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_3 + \left(t_0 + t_3 \cdot -2\right)\right)}{t_2}\\ \mathbf{elif}\;\phi_2 \leq 1.04 \cdot 10^{-106}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_4}{t_2}\\ \end{array} \]

Alternative 4
Error	3.9
Cost	84873

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 10^{-106}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 5
Error	7.3
Cost	71945

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.6 \cdot 10^{-11} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-11}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 6
Error	6.4
Cost	71680

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

Alternative 7
Error	7.4
Cost	59017

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{-10} \lor \neg \left(\phi_1 \leq 6.6 \cdot 10^{-12}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \phi_1 \cdot t_1}\\ \end{array} \]

Alternative 8
Error	11.0
Cost	58889

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4.9 \cdot 10^{-35} \lor \neg \left(\phi_2 \leq 9.8 \cdot 10^{-107}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{t_0 - \sin \phi_1 \cdot t_1}\\ \end{array} \]

Alternative 9
Error	17.0
Cost	52560

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -0.000155:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq -1.35 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{-228}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\lambda_2 \leq 2.85 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	12.8
Cost	52424

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{+23}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 11
Error	13.0
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -9600000000000 \lor \neg \left(\lambda_2 \leq 2.8 \cdot 10^{+23}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \end{array} \]

Alternative 12
Error	12.8
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{+23}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 13
Error	18.0
Cost	52297

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.45 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 2.8 \cdot 10^{+23}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 14
Error	12.6
Cost	52224

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

Alternative 15
Error	22.4
Cost	45960

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_1 - \sin \phi_1 \cdot t_0\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{+204}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 - \lambda_2 \cdot \cos \lambda_1}{t_2}\\ \mathbf{elif}\;\lambda_1 \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left|t_3\right|}{t_1 - \phi_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_3}{t_2}\\ \end{array} \]

Alternative 16
Error	21.1
Cost	45897

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -13800000000000 \lor \neg \left(\lambda_2 \leq 3.1 \cdot 10^{+23}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 17
Error	21.0
Cost	45833

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.00036 \lor \neg \left(\phi_2 \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 18
Error	21.0
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -2.65 \cdot 10^{+17}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.46 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 19
Error	20.9
Cost	45696

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

Alternative 20
Error	28.3
Cost	39433

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0004 \lor \neg \left(\phi_2 \leq 0.65\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 21
Error	22.7
Cost	39433

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 1.75 \cdot 10^{+37}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]

Alternative 22
Error	22.6
Cost	39432

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t_1\\ \mathbf{if}\;\phi_2 \leq -0.0004:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 0.029:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_2 \cdot \phi_1}\\ \end{array} \]

Alternative 23
Error	22.5
Cost	39432

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot t_2\\ \mathbf{if}\;\phi_2 \leq -0.0004:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \phi_1 \cdot t_1}\\ \mathbf{elif}\;\phi_2 \leq 0.027:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \lambda_2 \cdot \phi_1}\\ \end{array} \]

Alternative 24
Error	32.0
Cost	39168

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

Alternative 25
Error	32.5
Cost	32640

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

Reproduce

herbie shell --seed 2022340 
(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))))))