Average Error: 12.7 → 0.4
Time: 24.9s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sin \lambda_1}\\ \tan^{-1}_* \frac{\mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}\right)}^{2}, t_0 \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left({t_0}^{2}, t_0 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)} \end{array} \]
(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 lambda1))))
   (atan2
    (*
     (fma
      (pow (cbrt (* (sin lambda1) (cos lambda2))) 2.0)
      (* t_0 (cbrt (cos lambda2)))
      (- (* (cos lambda1) (sin lambda2))))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (* (cos phi2) (sin phi1))
      (fma
       (pow t_0 2.0)
       (* t_0 (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) {
	double t_0 = cbrt(sin(lambda1));
	return atan2((fma(pow(cbrt((sin(lambda1) * cos(lambda2))), 2.0), (t_0 * cbrt(cos(lambda2))), -(cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * fma(pow(t_0, 2.0), (t_0 * 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)
	t_0 = cbrt(sin(lambda1))
	return atan(Float64(fma((cbrt(Float64(sin(lambda1) * cos(lambda2))) ^ 2.0), Float64(t_0 * cbrt(cos(lambda2))), Float64(-Float64(cos(lambda1) * sin(lambda2)))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * fma((t_0 ^ 2.0), Float64(t_0 * 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_] := Block[{t$95$0 = N[Power[N[Sin[lambda1], $MachinePrecision], 1/3], $MachinePrecision]}, N[ArcTan[N[(N[(N[Power[N[Power[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 * N[Power[N[Cos[lambda2], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + (-N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(t$95$0 * N[Sin[lambda2], $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)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\sin \lambda_1}\\
\tan^{-1}_* \frac{\mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}\right)}^{2}, t_0 \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \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({t_0}^{2}, t_0 \cdot \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)}
\end{array}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 12.7

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

  2. Applied egg-rr6.5

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

  3. Applied egg-rr0.4

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1}\right)}^{2}, \sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)}} \]

  4. Applied egg-rr0.4

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \color{blue}{\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2}}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1}\right)}^{2}, \sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

  5. Applied egg-rr0.4

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}\right)}^{2}}, \sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1}\right)}^{2}, \sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

  6. Final simplification0.4

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}\right)}^{2}, \sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1}\right)}^{2}, \sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

Reproduce

herbie shell --seed 2022131 
(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))))))