Average Error: 0.1 → 0.2
Time: 23.1s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \sin \phi_1 \cdot \mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\cos delta}^{2}}{\cos delta + \log \left({\left(e^{\sin \phi_1}\right)}^{\left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)\right)}\right)} - \frac{{t_1}^{2}}{\cos delta + t_1}} \end{array} \]
(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
         (*
          (sin phi1)
          (fma
           (cos delta)
           (sin phi1)
           (* (sin delta) (* (cos phi1) (cos theta)))))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (-
      (/
       (pow (cos delta) 2.0)
       (+
        (cos delta)
        (log
         (pow
          (exp (sin phi1))
          (fma
           (cos delta)
           (sin phi1)
           (* (cos phi1) (* (sin delta) (cos theta))))))))
      (/ (pow t_1 2.0) (+ (cos delta) t_1)))))))
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 = sin(phi1) * fma(cos(delta), sin(phi1), (sin(delta) * (cos(phi1) * cos(theta))));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((pow(cos(delta), 2.0) / (cos(delta) + log(pow(exp(sin(phi1)), fma(cos(delta), sin(phi1), (cos(phi1) * (sin(delta) * cos(theta)))))))) - (pow(t_1, 2.0) / (cos(delta) + t_1))));
}

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(sin(phi1) * fma(cos(delta), sin(phi1), Float64(sin(delta) * Float64(cos(phi1) * cos(theta)))))
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(Float64((cos(delta) ^ 2.0) / Float64(cos(delta) + log((exp(sin(phi1)) ^ fma(cos(delta), sin(phi1), Float64(cos(phi1) * Float64(sin(delta) * cos(theta)))))))) - Float64((t_1 ^ 2.0) / Float64(cos(delta) + t_1)))))
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[Sin[phi1], $MachinePrecision] * N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Cos[delta], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] + N[Log[N[Power[N[Exp[N[Sin[phi1], $MachinePrecision]], $MachinePrecision], N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] + t$95$1), $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 := \sin \phi_1 \cdot \mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\cos delta}^{2}}{\cos delta + \log \left({\left(e^{\sin \phi_1}\right)}^{\left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)\right)}\right)} - \frac{{t_1}^{2}}{\cos delta + t_1}}
\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-rr0.1

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

  3. Applied egg-rr0.2

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

  4. Applied egg-rr0.2

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

  5. Final simplification0.2

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

Reproduce

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