Average Error: 12.9 → 0.2
Time: 50.6s
Precision: binary64
Cost: 110272
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1\right)} \]
(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 (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1))))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (* (cos phi2) (sin phi1))
    (+
     (log1p (expm1 (* (sin lambda1) (sin lambda2))))
     (* (cos lambda2) (cos 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(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * (log1p(expm1((sin(lambda1) * sin(lambda2)))) + (cos(lambda2) * cos(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(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(log1p(expm1(Float64(sin(lambda1) * sin(lambda2)))) + Float64(cos(lambda2) * cos(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[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[Log[1 + N[(Exp[N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[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(\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(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1\right)}

Error

Derivation

  1. Initial program 12.9

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

  2. Applied egg-rr12.7

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

  3. Applied egg-rr0.2

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

  4. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]
    Proof
    (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (*.f64 (neg.f64 (sin.f64 lambda2)) (cos.f64 lambda1))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (sin.f64 lambda2) (cos.f64 lambda1))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (sin.f64 lambda1) (cos.f64 lambda2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2))))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (sin.f64 lambda1) (cos.f64 lambda2)) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2))))): 10 points increase in error, 8 points decrease in error
    (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 lambda2) (sin.f64 lambda1))) (neg.f64 (*.f64 (cos.f64 lambda1) (sin.f64 lambda2)))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

  6. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

Alternatives

Alternative 1
Error0.2
Cost97472
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)} \]
Alternative 2
Error3.5
Cost91336
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -0.00012:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{-34}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\phi_2 \cdot \cos \phi_1 - t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error3.5
Cost91208
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -13:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{-34}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error0.2
Cost91136
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)} \]
Alternative 5
Error6.8
Cost71816
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.03:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error6.8
Cost71816
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := \tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot t_2}\\ \mathbf{if}\;\lambda_1 \leq -125:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 7
Error6.7
Cost71680
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 8
Error8.1
Cost65416
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -33500000000:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 540000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{\sqrt[3]{{t_0}^{3}} - \cos \phi_2 \cdot t_2}\\ \end{array} \]
Alternative 9
Error7.9
Cost65352
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -13500:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-29}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\sqrt[3]{{t_0}^{3}} - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]
Alternative 10
Error10.8
Cost65288
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.32 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.95:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1\right)\right)}\\ \end{array} \]
Alternative 11
Error10.8
Cost65288
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.85 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.95:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)\right)\right)}\\ \end{array} \]
Alternative 12
Error8.1
Cost65288
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.8 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 4.3 \cdot 10^{-54}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right) - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]
Alternative 13
Error8.1
Cost65288
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -13500:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 4.3 \cdot 10^{-54}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right) - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]
Alternative 14
Error19.5
Cost52624
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot t_0\\ t_2 := \sin \lambda_1 \cdot \cos \phi_2\\ t_3 := \cos \phi_1 \cdot \sin \phi_2\\ t_4 := \tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{t_3 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -9.5 \cdot 10^{-73}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\lambda_2 \leq 3.25 \cdot 10^{-183}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - \cos \phi_2 \cdot t_1}\\ \mathbf{elif}\;\lambda_2 \leq 10^{-97}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_3 - t_1}\\ \mathbf{elif}\;\lambda_2 \leq 5.1 \cdot 10^{-57}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]
Alternative 15
Error19.5
Cost52560
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ t_2 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -1.1 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 3.1 \cdot 10^{-182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 9.4 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 16
Error19.5
Cost52560
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \lambda_1 \cdot \cos \phi_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{t_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -8 \cdot 10^{-72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_2 \leq 4 \cdot 10^{-183}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \phi_2 \cdot t_0}\\ \mathbf{elif}\;\lambda_2 \leq 1.25 \cdot 10^{-102}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_2 - t_0}\\ \mathbf{elif}\;\lambda_2 \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 17
Error21.8
Cost52496
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1}\\ t_3 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{-183}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_2 \leq 5.5 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2 - \cos \phi_2 \cdot t_1}\\ \end{array} \]
Alternative 18
Error23.1
Cost52488
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \cos \phi_2 \cdot t_0}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 10^{-75}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - t_0}\\ \end{array} \]
Alternative 19
Error10.8
Cost52488
\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.95:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 20
Error10.8
Cost52488
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.95:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]
Alternative 21
Error13.5
Cost52360
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.55 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 22
Error13.1
Cost52360
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -125:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2}\\ \mathbf{elif}\;\lambda_1 \leq 0.00044:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_2}\\ \end{array} \]
Alternative 23
Error21.8
Cost45696
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
Alternative 24
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 25
Error22.4
Cost39168
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)} \]
Alternative 26
Error33.2
Cost32840
\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.55 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 27
Error35.0
Cost32776
\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\lambda_2 \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 7.8 \cdot 10^{-202}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 28
Error33.2
Cost32776
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 0.0004:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
Alternative 29
Error32.7
Cost32640
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)} \]
Alternative 30
Error32.7
Cost26504
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-29}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 31
Error33.4
Cost26440
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 8.2 \cdot 10^{-32}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 32
Error43.1
Cost19456
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

Error

Reproduce

herbie shell --seed 2022331 
(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))))))