Average Error: 0.2 → 0.2
Time: 21.3s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(\cos \phi_1, \sin delta \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\\ t_2 := \cos delta + \sin \phi_1 \cdot t_1\\ \lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin delta \cdot \sin theta\right)\right) \cdot \cos \phi_1}{\frac{{\cos delta}^{2}}{t_2} - \frac{{t_1}^{2} \cdot {\sin \phi_1}^{2}}{t_2}} \end{array} \]
(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 phi1)
          (* (sin delta) (cos theta))
          (* (cos delta) (sin phi1))))
        (t_2 (+ (cos delta) (* (sin phi1) t_1))))
   (+
    lambda1
    (atan2
     (* (log1p (expm1 (* (sin delta) (sin theta)))) (cos phi1))
     (-
      (/ (pow (cos delta) 2.0) t_2)
      (/ (* (pow t_1 2.0) (pow (sin phi1) 2.0)) 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 = fma(cos(phi1), (sin(delta) * cos(theta)), (cos(delta) * sin(phi1)));
	double t_2 = cos(delta) + (sin(phi1) * t_1);
	return lambda1 + atan2((log1p(expm1((sin(delta) * sin(theta)))) * cos(phi1)), ((pow(cos(delta), 2.0) / t_2) - ((pow(t_1, 2.0) * pow(sin(phi1), 2.0)) / 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 = fma(cos(phi1), Float64(sin(delta) * cos(theta)), Float64(cos(delta) * sin(phi1)))
	t_2 = Float64(cos(delta) + Float64(sin(phi1) * t_1))
	return Float64(lambda1 + atan(Float64(log1p(expm1(Float64(sin(delta) * sin(theta)))) * cos(phi1)), Float64(Float64((cos(delta) ^ 2.0) / t_2) - Float64(Float64((t_1 ^ 2.0) * (sin(phi1) ^ 2.0)) / 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[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[delta], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Log[1 + N[(Exp[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Cos[delta], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $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 \phi_1, \sin delta \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\\
t_2 := \cos delta + \sin \phi_1 \cdot t_1\\
\lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin delta \cdot \sin theta\right)\right) \cdot \cos \phi_1}{\frac{{\cos delta}^{2}}{t_2} - \frac{{t_1}^{2} \cdot {\sin \phi_1}^{2}}{t_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 0.2

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

  2. Applied egg-rr0.2

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

  3. Applied egg-rr0.2

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

  4. Applied egg-rr0.2

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

  5. Applied egg-rr0.2

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

  6. Final simplification0.2

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

Reproduce

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