Average Error: 13.2 → 0.2
Time: 46.9s
Precision: binary64
Cost: 136512
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \sin \phi_1}{\frac{t_0 - t_1}{{t_0}^{2} - {t_1}^{2}}}} \end{array} \]
(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
 (let* ((t_0 (* (cos lambda2) (cos lambda1)))
        (t_1 (* (sin lambda1) (sin lambda2))))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1))))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (/
      (* (cos phi2) (sin phi1))
      (/ (- t_0 t_1) (- (pow t_0 2.0) (pow t_1 2.0))))))))
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) {
	double t_0 = cos(lambda2) * cos(lambda1);
	double t_1 = sin(lambda1) * sin(lambda2);
	return atan2((fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) / ((t_0 - t_1) / (pow(t_0, 2.0) - pow(t_1, 2.0))))));
}

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)
	t_0 = Float64(cos(lambda2) * cos(lambda1))
	t_1 = Float64(sin(lambda1) * sin(lambda2))
	return atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1)))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) / Float64(Float64(t_0 - t_1) / Float64((t_0 ^ 2.0) - (t_1 ^ 2.0))))))
end

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_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 - t$95$1), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

\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)}
↓
\begin{array}{l}
t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\
t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\
\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \sin \phi_1}{\frac{t_0 - t_1}{{t_0}^{2} - {t_1}^{2}}}}
\end{array}

Error

Derivation

  1. Initial program 13.2

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

  2. Applied egg-rr6.8

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

  3. Applied egg-rr0.2

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

  4. Simplified0.2

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

    [Start]0.2

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

    associate-/l* [=>]0.2

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

    *-commutative [<=]0.2

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

    associate-*l* [=>]0.2

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

  5. Taylor expanded in lambda1 around inf 0.2

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

  6. Simplified0.2

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

    [Start]0.2

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

    *-commutative [=>]0.2

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

    unpow2 [=>]0.2

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

    unpow2 [=>]0.2

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

    swap-sqr [<=]0.2

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

    unpow2 [<=]0.2

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

    *-commutative [<=]0.2

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

    *-commutative [=>]0.2

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

    unpow2 [=>]0.2

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

    unpow2 [=>]0.2

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

    swap-sqr [<=]0.2

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

    unpow2 [<=]0.2

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

    *-commutative [=>]0.2

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

  7. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost103744
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]
Alternative 2
Error0.2
Cost97472
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \]
Alternative 3
Error3.7
Cost84873
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-12} \lor \neg \left(\phi_2 \leq 2.2 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]
Alternative 4
Error6.8
Cost78016
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 5
Error7.7
Cost71945
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 1.12 \cdot 10^{-8}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
Alternative 6
Error7.8
Cost71752
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \sqrt[3]{{t_1}^{3}}}\\ \mathbf{elif}\;\phi_1 \leq 5.3 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot t_1}\\ \end{array} \]
Alternative 7
Error6.8
Cost71680
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 8
Error7.8
Cost65220
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \sqrt[3]{{t_1}^{3}}}\\ \mathbf{elif}\;\phi_1 \leq 1.22 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot t_1}\\ \end{array} \]
Alternative 9
Error7.8
Cost65156
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_3\right)\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_3}{t_0 - \cos \phi_2 \cdot t_2}\\ \end{array} \]
Alternative 10
Error7.8
Cost58888
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \mathbf{elif}\;\phi_1 \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \phi_2 \cdot t_2}\\ \end{array} \]
Alternative 11
Error13.5
Cost52425
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq -29500000000000 \lor \neg \left(\lambda_2 \leq 1.4 \cdot 10^{+14}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \end{array} \]
Alternative 12
Error13.5
Cost52360
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \sin \lambda_1 \cdot \cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.00038:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_1 \cdot t_0}\\ \end{array} \]
Alternative 13
Error13.4
Cost52360
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2}\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_2}\\ \end{array} \]
Alternative 14
Error19.2
Cost52233
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{+91} \lor \neg \left(\lambda_1 \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
Alternative 15
Error19.2
Cost52232
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \lambda_1 \cdot \cos \phi_2\\ t_3 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.95 \cdot 10^{+91}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - t_0 \cdot t_1}\\ \mathbf{elif}\;\lambda_1 \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_3 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - \cos \lambda_1 \cdot t_0}\\ \end{array} \]
Alternative 16
Error13.2
Cost52224
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
Alternative 17
Error19.8
Cost46345
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.0002 \lor \neg \left(\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-59}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]
Alternative 18
Error22.2
Cost45832
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -1.15 \cdot 10^{+26}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2 - t_0 \cdot t_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.45 \cdot 10^{-131}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\sin \phi_2 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\sin \phi_2 - \sin \phi_1 \cdot t_1}\\ \end{array} \]
Alternative 19
Error21.8
Cost45696
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 20
Error21.7
Cost45696
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 21
Error31.8
Cost39177
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -34000000000 \lor \neg \left(\phi_1 \leq 7.6 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \end{array} \]
Alternative 22
Error22.4
Cost39168
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 23
Error32.5
Cost32640
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 24
Error33.3
Cost26441
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{-39} \lor \neg \left(\phi_1 \leq 6 \cdot 10^{-6}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \end{array} \]
Alternative 25
Error33.1
Cost26372
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \end{array} \]
Alternative 26
Error45.0
Cost19657
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.45 \cdot 10^{-115} \lor \neg \left(\lambda_2 \leq 3.5 \cdot 10^{-72}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \end{array} \]
Alternative 27
Error43.4
Cost19456
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]
Alternative 28
Error48.5
Cost19328
\[\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2} \]

Error

Reproduce

herbie shell --seed 2022364 
(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))))))