Average Error: 0.2 → 0.2
Time: 21.8s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \cos delta \cdot \sin \phi_1\\ t_2 := \sin \phi_1 \cdot \mathsf{fma}\left(\cos \phi_1, \sin delta \cdot \cos theta, t_1\right)\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{{\cos delta}^{3} - {\left(t_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}{\cos delta \cdot \cos delta + \left(t_2 \cdot t_2 + \cos delta \cdot t_2\right)}} \end{array} \]
(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 (* (cos delta) (sin phi1)))
        (t_2 (* (sin phi1) (fma (cos phi1) (* (sin delta) (cos theta)) t_1))))
   (+
    lambda1
    (atan2
     (* (sin theta) (* (sin delta) (cos phi1)))
     (/
      (-
       (pow (cos delta) 3.0)
       (*
        (pow (+ t_1 (* (sin delta) (* (cos phi1) (cos theta)))) 3.0)
        (pow (sin phi1) 3.0)))
      (+ (* (cos delta) (cos delta)) (+ (* t_2 t_2) (* (cos delta) t_2))))))))
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 = cos(delta) * sin(phi1);
	double t_2 = sin(phi1) * fma(cos(phi1), (sin(delta) * cos(theta)), t_1);
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), ((pow(cos(delta), 3.0) - (pow((t_1 + (sin(delta) * (cos(phi1) * cos(theta)))), 3.0) * pow(sin(phi1), 3.0))) / ((cos(delta) * cos(delta)) + ((t_2 * t_2) + (cos(delta) * t_2)))));
}

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 = Float64(cos(delta) * sin(phi1))
	t_2 = Float64(sin(phi1) * fma(cos(phi1), Float64(sin(delta) * cos(theta)), t_1))
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(Float64((cos(delta) ^ 3.0) - Float64((Float64(t_1 + Float64(sin(delta) * Float64(cos(phi1) * cos(theta)))) ^ 3.0) * (sin(phi1) ^ 3.0))) / Float64(Float64(cos(delta) * cos(delta)) + Float64(Float64(t_2 * t_2) + Float64(cos(delta) * t_2))))))
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]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Cos[delta], $MachinePrecision], 3.0], $MachinePrecision] - N[(N[Power[N[(t$95$1 + N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Sin[phi1], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[delta], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * t$95$2), $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 := \cos delta \cdot \sin \phi_1\\
t_2 := \sin \phi_1 \cdot \mathsf{fma}\left(\cos \phi_1, \sin delta \cdot \cos theta, t_1\right)\\
\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{{\cos delta}^{3} - {\left(t_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}{\cos delta \cdot \cos delta + \left(t_2 \cdot t_2 + \cos delta \cdot t_2\right)}}
\end{array}

Error

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Applied egg-rr0.2

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

  4. Taylor expanded in phi1 around inf 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022203 
(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))))))))))