Average Error: 13.3 → 0.2
Time: 22.1s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \sqrt[3]{{\left(\sin \lambda_1 \cdot \left(\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}^{3}}\right)} \]
(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
 (atan2
  (log1p
   (expm1
    (*
     (cos phi2)
     (- (* (cos lambda2) (sin lambda1)) (* (sin lambda2) (cos lambda1))))))
  (-
   (* (cos phi1) (sin phi2))
   (+
    (* (* (cos phi2) (sin phi1)) (* (cos lambda2) (cos lambda1)))
    (cbrt
     (pow
      (* (sin lambda1) (* (sin phi1) (* (cos phi2) (sin lambda2))))
      3.0))))))
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) {
	return atan2(log1p(expm1((cos(phi2) * ((cos(lambda2) * sin(lambda1)) - (sin(lambda2) * cos(lambda1)))))), ((cos(phi1) * sin(phi2)) - (((cos(phi2) * sin(phi1)) * (cos(lambda2) * cos(lambda1))) + cbrt(pow((sin(lambda1) * (sin(phi1) * (cos(phi2) * sin(lambda2)))), 3.0)))));
}

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}
↓
public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.log1p(Math.expm1((Math.cos(phi2) * ((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.sin(lambda2) * Math.cos(lambda1)))))), ((Math.cos(phi1) * Math.sin(phi2)) - (((Math.cos(phi2) * Math.sin(phi1)) * (Math.cos(lambda2) * Math.cos(lambda1))) + Math.cbrt(Math.pow((Math.sin(lambda1) * (Math.sin(phi1) * (Math.cos(phi2) * Math.sin(lambda2)))), 3.0)))));
}

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)
	return atan(log1p(expm1(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(sin(lambda2) * cos(lambda1)))))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(Float64(cos(phi2) * sin(phi1)) * Float64(cos(lambda2) * cos(lambda1))) + cbrt((Float64(sin(lambda1) * Float64(sin(phi1) * Float64(cos(phi2) * sin(lambda2)))) ^ 3.0)))))
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_] := N[ArcTan[N[Log[1 + N[(Exp[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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)}
↓
\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)\right)\right)}{\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) + \sqrt[3]{{\left(\sin \lambda_1 \cdot \left(\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)}^{3}}\right)}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.3

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

  2. Applied sin-diff_binary647.0

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

  3. Applied cos-diff_binary640.2

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

  4. Applied distribute-lft-in_binary640.2

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

  5. Applied log1p-expm1-u_binary640.2

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

  6. Simplified0.2

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

  7. Applied add-cbrt-cube_binary640.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2022137 
(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))))))