Average Error: 13.1 → 0.2
Time: 27.5s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1}\\ \tan^{-1}_* \frac{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) + \mathsf{fma}\left(-t_0, t_0 \cdot \left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2}\right), t_0 \cdot \left(t_0 \cdot t_0\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \end{array} \]
(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 (cbrt (* (sin lambda2) (cos lambda1)))))
   (atan2
    (*
     (+
      (fma
       1.0
       (* (cos lambda2) (sin lambda1))
       (* (sin lambda2) (- (cos lambda1))))
      (fma
       (- t_0)
       (* t_0 (* (cbrt (cos lambda1)) (cbrt (sin lambda2))))
       (* t_0 (* t_0 t_0))))
     (cos phi2))
    (fma
     (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))
     (* (cos phi2) (- (sin phi1)))
     (* (cos phi1) (sin phi2))))))
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 = cbrt((sin(lambda2) * cos(lambda1)));
	return atan2(((fma(1.0, (cos(lambda2) * sin(lambda1)), (sin(lambda2) * -cos(lambda1))) + fma(-t_0, (t_0 * (cbrt(cos(lambda1)) * cbrt(sin(lambda2)))), (t_0 * (t_0 * t_0)))) * cos(phi2)), fma(fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))), (cos(phi2) * -sin(phi1)), (cos(phi1) * sin(phi2))));
}

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 = cbrt(Float64(sin(lambda2) * cos(lambda1)))
	return atan(Float64(Float64(fma(1.0, Float64(cos(lambda2) * sin(lambda1)), Float64(sin(lambda2) * Float64(-cos(lambda1)))) + fma(Float64(-t_0), Float64(t_0 * Float64(cbrt(cos(lambda1)) * cbrt(sin(lambda2)))), Float64(t_0 * Float64(t_0 * t_0)))) * cos(phi2)), fma(fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))), Float64(cos(phi2) * Float64(-sin(phi1))), Float64(cos(phi1) * sin(phi2))))
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[Power[N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[ArcTan[N[(N[(N[(1.0 * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[((-t$95$0) * N[(t$95$0 * N[(N[Power[N[Cos[lambda1], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[lambda2], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $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 := \sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1}\\
\tan^{-1}_* \frac{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) + \mathsf{fma}\left(-t_0, t_0 \cdot \left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2}\right), t_0 \cdot \left(t_0 \cdot t_0\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)}
\end{array}

Error

Derivation

  1. Initial program 13.1

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

  2. Simplified13.1

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

  3. Applied egg-rr6.8

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, -\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\right)\right)\right)} \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

  4. Applied egg-rr0.4

    \[\leadsto \tan^{-1}_* \frac{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, -\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

  5. Taylor expanded in lambda1 around inf 0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, -\color{blue}{\sin \lambda_2 \cdot \cos \lambda_1}\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

  6. Applied egg-rr0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, -\sin \lambda_2 \cdot \cos \lambda_1\right) + \mathsf{fma}\left(-\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \color{blue}{\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2}\right)}, \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

  7. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\mathsf{fma}\left(1, \cos \lambda_2 \cdot \sin \lambda_1, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1}, \sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1} \cdot \left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2}\right), \sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \cos \lambda_1}\right)\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

Reproduce

herbie shell --seed 2022185 
(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))))))