?

Average Error: 13.4 → 0.2
Time: 53.4s
Precision: binary64
Cost: 103744

?

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]
(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 (cos lambda1) (- (sin lambda2)) (* (sin lambda1) (cos lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (* (cos phi2) (sin phi1))
    (fma (cos lambda1) (cos lambda2) (* (sin lambda2) (sin lambda1)))))))
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(cos(lambda1), -sin(lambda2), (sin(lambda1) * cos(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * fma(cos(lambda1), cos(lambda2), (sin(lambda2) * sin(lambda1))))));
}

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(cos(lambda1), Float64(-sin(lambda2)), Float64(sin(lambda1) * cos(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda2) * sin(lambda1))))))
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_] := N[ArcTan[N[(N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision]) + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $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)}
↓
\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\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(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)}

Error?

Derivation?

  1. Initial program 13.4

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

  2. Simplified13.4

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

    [Start]13.4

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

    associate-*l* [=>]13.4

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

  3. Applied egg-rr6.8

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

  4. Simplified6.8

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

    [Start]6.8

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

    +-commutative [<=]6.8

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

    distribute-lft-neg-in [<=]6.8

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

    distribute-rgt-neg-in [=>]6.8

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

    sin-neg [<=]6.8

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

    fma-def [=>]6.8

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

    sin-neg [=>]6.8

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

  5. Applied egg-rr0.2

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

  6. Simplified0.2

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

    [Start]0.2

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

    distribute-lft-out [=>]0.2

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

    *-commutative [<=]0.2

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

    fma-def [=>]0.2

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

  7. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost97472
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \]
Alternative 2
Error0.2
Cost97408
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)} \]
Alternative 3
Error3.9
Cost91208
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_3 := \mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, t_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -3300000:\\ \;\;\;\;\tan^{-1}_* \frac{t_2 \cdot \cos \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \lambda_2\right)}{t_1}\\ \mathbf{elif}\;\phi_2 \leq 10^{-37}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \sin \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_1}\\ \end{array} \]
Alternative 4
Error3.8
Cost91144
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_3 := \sin \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -0.0056:\\ \;\;\;\;\tan^{-1}_* \frac{t_3 \cdot \cos \phi_2 - \cos \phi_2 \cdot t_0}{t_2}\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-37}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3 - t_0}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, t_3\right) \cdot \cos \phi_2}{t_2}\\ \end{array} \]
Alternative 5
Error0.2
Cost91136
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \]
Alternative 6
Error3.8
Cost84872
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_3 := \sin \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -0.0056:\\ \;\;\;\;\tan^{-1}_* \frac{t_3 \cdot \cos \phi_2 - \cos \phi_2 \cdot t_0}{t_2}\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-37}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3 - t_0}{t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, t_3\right) \cdot \cos \phi_2}{t_2}\\ \end{array} \]
Alternative 7
Error6.8
Cost78016
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
Alternative 8
Error7.9
Cost71945
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 5100\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]
Alternative 9
Error6.8
Cost71680
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
Alternative 10
Error8.0
Cost71624
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 + \frac{\sin \phi_1}{\frac{-1}{\cos \phi_2 \cdot t_2}}}\\ \mathbf{elif}\;\phi_1 \leq 5100:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)\right)}\\ \end{array} \]
Alternative 11
Error7.9
Cost65416
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 + \frac{\sin \phi_1}{\frac{-1}{t_3}}}\\ \mathbf{elif}\;\phi_1 \leq 5100:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)\right)}\\ \end{array} \]
Alternative 12
Error7.9
Cost65288
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 + \frac{\sin \phi_1}{\frac{-1}{\cos \phi_2 \cdot t_2}}}\\ \mathbf{elif}\;\phi_1 \leq 5100:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)\right)}\\ \end{array} \]
Alternative 13
Error7.9
Cost65224
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -5.4 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 + \frac{\sin \phi_1}{\frac{-1}{t_3}}}\\ \mathbf{elif}\;\phi_1 \leq 5100:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot t_3}\\ \end{array} \]
Alternative 14
Error7.9
Cost59016
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 + \frac{\sin \phi_1}{\frac{-1}{t_3}}}\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot t_3}\\ \end{array} \]
Alternative 15
Error11.2
Cost58888
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0056:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \mathbf{elif}\;\phi_2 \leq 0.0092:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \end{array} \]
Alternative 16
Error13.5
Cost52424
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.028:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 0.185:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
Alternative 17
Error13.6
Cost52361
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -135000 \lor \neg \left(\lambda_1 \leq 95\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
Alternative 18
Error13.6
Cost52361
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -135000 \lor \neg \left(\lambda_1 \leq 85\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \end{array} \]
Alternative 19
Error13.8
Cost52360
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -135000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\lambda_1 \leq 2.8 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
Alternative 20
Error13.4
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 21
Error13.4
Cost52224
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
Alternative 22
Error20.1
Cost46345
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.4 \lor \neg \left(\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-53}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \end{array} \]
Alternative 23
Error22.2
Cost45896
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.028:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 0.185:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
Alternative 24
Error22.2
Cost45896
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.031:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 0.26:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
Alternative 25
Error29.8
Cost45833
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.22 \cdot 10^{-29} \lor \neg \left(\lambda_1 \leq 2.15 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]
Alternative 26
Error22.3
Cost45833
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -160000 \lor \neg \left(\lambda_1 \leq 8 \cdot 10^{+39}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]
Alternative 27
Error30.2
Cost45700
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]
Alternative 28
Error22.2
Cost45696
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
Alternative 29
Error22.1
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 30
Error25.6
Cost45568
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1} \]
Alternative 31
Error39.6
Cost39432
\[\begin{array}{l} t_0 := \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \lambda_1 \cdot \cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2 - t_0\\ \mathbf{if}\;\phi_2 \leq -1.45:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot -0.5\right) + 1\right) - t_0}\\ \mathbf{elif}\;\phi_2 \leq 1.32 \cdot 10^{+35}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2}\\ \end{array} \]
Alternative 32
Error31.1
Cost39428
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -260000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]
Alternative 33
Error32.2
Cost39364
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1250000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \left(-\cos \phi_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]
Alternative 34
Error33.2
Cost39300
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -500000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot -0.5\right) + 1\right) - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]
Alternative 35
Error39.6
Cost33284
\[\begin{array}{l} t_0 := \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \lambda_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.115:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot -0.5\right) + 1\right) - t_0}\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{+34}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0}\\ \end{array} \]
Alternative 36
Error39.6
Cost33033
\[\begin{array}{l} t_0 := \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \lor \neg \left(\phi_2 \leq 1.1 \cdot 10^{+35}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - t_0}\\ \end{array} \]
Alternative 37
Error49.4
Cost32640
\[\tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 38
Error44.4
Cost32640
\[\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

Error

Reproduce?

herbie shell --seed 2023073 
(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))))))