Average Error: 13.1 → 0.2
Time: 21.0s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\ \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{{t_0}^{3} + {t_1}^{3}}{\mathsf{log1p}\left(\mathsf{expm1}\left({t_0}^{2}\right)\right) + \left({\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2} - t_0 \cdot t_1\right)}, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \end{array} \]
(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 (* (cos lambda2) (cos lambda1)))
        (t_1 (* (sin lambda1) (sin lambda2))))
   (atan2
    (*
     (- (* (cos lambda2) (sin lambda1)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (fma
     (/
      (+ (pow t_0 3.0) (pow t_1 3.0))
      (+
       (log1p (expm1 (pow t_0 2.0)))
       (- (* (pow (sin lambda1) 2.0) (pow (sin lambda2) 2.0)) (* t_0 t_1))))
     (* (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 = sin(lambda1) * sin(lambda2);
	return atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), fma(((pow(t_0, 3.0) + pow(t_1, 3.0)) / (log1p(expm1(pow(t_0, 2.0))) + ((pow(sin(lambda1), 2.0) * pow(sin(lambda2), 2.0)) - (t_0 * t_1)))), (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 = Float64(sin(lambda1) * sin(lambda2))
	return atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), fma(Float64(Float64((t_0 ^ 3.0) + (t_1 ^ 3.0)) / Float64(log1p(expm1((t_0 ^ 2.0))) + Float64(Float64((sin(lambda1) ^ 2.0) * (sin(lambda2) ^ 2.0)) - Float64(t_0 * t_1)))), 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[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Log[1 + N[(Exp[N[Power[t$95$0, 2.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Power[N[Sin[lambda1], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[lambda2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $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 := \cos \lambda_2 \cdot \cos \lambda_1\\
t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\
\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{{t_0}^{3} + {t_1}^{3}}{\mathsf{log1p}\left(\mathsf{expm1}\left({t_0}^{2}\right)\right) + \left({\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2} - t_0 \cdot t_1\right)}, \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 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-rr6.7

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

  4. Applied egg-rr0.2

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

  5. Taylor expanded in lambda1 around inf 0.2

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

  6. Applied egg-rr0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{2}\right)\right)} + \left({\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2} - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

  7. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\frac{{\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^{2}\right)\right) + \left({\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2} - \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}, \cos \phi_2 \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)} \]

Reproduce

herbie shell --seed 2022150 
(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))))))