Average Error: 0.1 → 0.2
Time: 25.3s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{1}{\frac{\cos delta + \sin \phi_1 \cdot t_1}{{\cos delta}^{2} - {\log \left({\left(e^{\sin \phi_1}\right)}^{t_1}\right)}^{2}}}} \end{array} \]
(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (sin phi1) (cos delta))
        (* (* (cos phi1) (sin delta)) (cos theta))))))))))
↓
(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (fma
          (cos delta)
          (sin phi1)
          (* (* (sin delta) (cos phi1)) (cos theta)))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (/
      1.0
      (/
       (+ (cos delta) (* (sin phi1) t_1))
       (-
        (pow (cos delta) 2.0)
        (pow (log (pow (exp (sin phi1)) t_1)) 2.0))))))))
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
}
↓
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = fma(cos(delta), sin(phi1), ((sin(delta) * cos(phi1)) * cos(theta)));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0 / ((cos(delta) + (sin(phi1) * t_1)) / (pow(cos(delta), 2.0) - pow(log(pow(exp(sin(phi1)), t_1)), 2.0)))));
}

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)))))))))
end
↓
function code(lambda1, phi1, phi2, delta, theta)
	t_1 = fma(cos(delta), sin(phi1), Float64(Float64(sin(delta) * cos(phi1)) * cos(theta)))
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(1.0 / Float64(Float64(cos(delta) + Float64(sin(phi1) * t_1)) / Float64((cos(delta) ^ 2.0) - (log((exp(sin(phi1)) ^ t_1)) ^ 2.0))))))
end

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[Cos[delta], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Cos[delta], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[N[Power[N[Exp[N[Sin[phi1], $MachinePrecision]], $MachinePrecision], t$95$1], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}
↓
\begin{array}{l}
t_1 := \mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{1}{\frac{\cos delta + \sin \phi_1 \cdot t_1}{{\cos delta}^{2} - {\log \left({\left(e^{\sin \phi_1}\right)}^{t_1}\right)}^{2}}}}
\end{array}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

  1. Initial program 0.1

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

  2. Applied egg-rr0.2

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

  3. Applied egg-rr0.2

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

  4. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{1}{\frac{\cos delta + \sin \phi_1 \cdot \mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)}{{\cos delta}^{2} - {\log \left({\left(e^{\sin \phi_1}\right)}^{\left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)}\right)}^{2}}}} \]

Reproduce

herbie shell --seed 2022146 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))