Average Error: 13.1 → 0.4
Time: 24.6s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\\ \tan^{-1}_* \frac{\mathsf{fma}\left(t_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right), \sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \end{array} \]
(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 (* (cos phi2) (sin phi1)))
        (t_1 (cbrt (* (cos lambda2) (sin lambda1)))))
   (atan2
    (*
     (fma
      (* t_1 (log1p (expm1 t_1)))
      (* (cbrt (sin lambda1)) (cbrt (cos lambda2)))
      (- (* (cos lambda1) (sin lambda2))))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (+
      (* t_0 (* (cos lambda2) (cos lambda1)))
      (* t_0 (* (sin lambda1) (sin lambda2))))))))
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(phi2) * sin(phi1);
	double t_1 = cbrt((cos(lambda2) * sin(lambda1)));
	return atan2((fma((t_1 * log1p(expm1(t_1))), (cbrt(sin(lambda1)) * cbrt(cos(lambda2))), -(cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((t_0 * (cos(lambda2) * cos(lambda1))) + (t_0 * (sin(lambda1) * sin(lambda2))))));
}

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(phi2) * sin(phi1))
	t_1 = cbrt(Float64(cos(lambda2) * sin(lambda1)))
	return atan(Float64(fma(Float64(t_1 * log1p(expm1(t_1))), Float64(cbrt(sin(lambda1)) * cbrt(cos(lambda2))), Float64(-Float64(cos(lambda1) * sin(lambda2)))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(t_0 * Float64(cos(lambda2) * cos(lambda1))) + Float64(t_0 * Float64(sin(lambda1) * sin(lambda2))))))
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[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[ArcTan[N[(N[(N[(t$95$1 * N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[lambda1], $MachinePrecision], 1/3], $MachinePrecision] * 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[(t$95$0 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $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 \phi_2 \cdot \sin \phi_1\\
t_1 := \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\\
\tan^{-1}_* \frac{\mathsf{fma}\left(t_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right), \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(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}
\end{array}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

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. Applied egg-rr6.9

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

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.4

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

  5. Applied egg-rr0.4

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\right)\right)}, \sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]

  6. Final simplification0.4

    \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\right)\right), \sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2}, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]

Reproduce

herbie shell --seed 2022145 
(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))))))