Average Error: 12.9 → 0.3
Time: 26.8s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\\ t_2 := t_1 \cdot t_1\\ t_3 := \sin \lambda_1 \cdot \sin \lambda_2\\ \tan^{-1}_* \frac{\left(\mathsf{fma}\left(\cos \lambda_2, \sin \lambda_1, t_1 \cdot \left(-t_2\right)\right) + \mathsf{fma}\left(-t_1, t_1 \cdot \left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2}\right), t_1 \cdot t_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{t_0 \cdot t_0 - t_3 \cdot t_3}{t_0 - t_3}, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \end{array} \]
(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 (* (cos lambda2) (cos lambda1)))
        (t_1 (cbrt (* (cos lambda1) (sin lambda2))))
        (t_2 (* t_1 t_1))
        (t_3 (* (sin lambda1) (sin lambda2))))
   (atan2
    (*
     (+
      (fma (cos lambda2) (sin lambda1) (* t_1 (- t_2)))
      (fma
       (- t_1)
       (* t_1 (* (cbrt (cos lambda1)) (cbrt (sin lambda2))))
       (* t_1 t_2)))
     (cos phi2))
    (fma
     (/ (- (* t_0 t_0) (* t_3 t_3)) (- t_0 t_3))
     (* (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 = cos(lambda2) * cos(lambda1);
	double t_1 = cbrt((cos(lambda1) * sin(lambda2)));
	double t_2 = t_1 * t_1;
	double t_3 = sin(lambda1) * sin(lambda2);
	return atan2(((fma(cos(lambda2), sin(lambda1), (t_1 * -t_2)) + fma(-t_1, (t_1 * (cbrt(cos(lambda1)) * cbrt(sin(lambda2)))), (t_1 * t_2))) * cos(phi2)), fma((((t_0 * t_0) - (t_3 * t_3)) / (t_0 - t_3)), (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 = Float64(cos(lambda2) * cos(lambda1))
	t_1 = cbrt(Float64(cos(lambda1) * sin(lambda2)))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(sin(lambda1) * sin(lambda2))
	return atan(Float64(Float64(fma(cos(lambda2), sin(lambda1), Float64(t_1 * Float64(-t_2))) + fma(Float64(-t_1), Float64(t_1 * Float64(cbrt(cos(lambda1)) * cbrt(sin(lambda2)))), Float64(t_1 * t_2))) * cos(phi2)), fma(Float64(Float64(Float64(t_0 * t_0) - Float64(t_3 * t_3)) / Float64(t_0 - t_3)), 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[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(t$95$1 * (-t$95$2)), $MachinePrecision]), $MachinePrecision] + N[((-t$95$1) * N[(t$95$1 * N[(N[Power[N[Cos[lambda1], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[lambda2], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$3), $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 := \cos \lambda_2 \cdot \cos \lambda_1\\
t_1 := \sqrt[3]{\cos \lambda_1 \cdot \sin \lambda_2}\\
t_2 := t_1 \cdot t_1\\
t_3 := \sin \lambda_1 \cdot \sin \lambda_2\\
\tan^{-1}_* \frac{\left(\mathsf{fma}\left(\cos \lambda_2, \sin \lambda_1, t_1 \cdot \left(-t_2\right)\right) + \mathsf{fma}\left(-t_1, t_1 \cdot \left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\sin \lambda_2}\right), t_1 \cdot t_2\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{t_0 \cdot t_0 - t_3 \cdot t_3}{t_0 - t_3}, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)}
\end{array}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 12.9

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

  2. Simplified12.9

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

  3. Applied egg-rr6.7

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

  4. Applied egg-rr0.3

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

  5. Applied egg-rr0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2022165 
(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))))))