Destination given bearing on a great circle

Average Error: 99.7% → 99.7%

Time: 27.0s

Precision: binary64

Cost: 109888.00

Math

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

↓

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

FPCore

(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
 (+
  lambda1
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (fma
    (sin
     (log
      (exp
       (asin
        (fma
         (sin delta)
         (* (cos phi1) (cos theta))
         (* (cos delta) (sin phi1)))))))
    (- (sin phi1))
    (cos delta)))))

C

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) {
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), fma(sin(log(exp(asin(fma(sin(delta), (cos(phi1) * cos(theta)), (cos(delta) * sin(phi1))))))), -sin(phi1), cos(delta)));
}

Julia

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)
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), fma(sin(log(exp(asin(fma(sin(delta), Float64(cos(phi1) * cos(theta)), Float64(cos(delta) * sin(phi1))))))), Float64(-sin(phi1)), cos(delta))))
end

Wolfram

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_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[Log[N[Exp[N[ArcSin[N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision]) + N[Cos[delta], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7

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

2. Simplified 99.8

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

   [Start] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   associate-*l* [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   cancel-sign-sub-inv [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta + \left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]
   +-commutative [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) + \cos delta}} \]
   *-commutative [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \left(-\sin \phi_1\right)} + \cos delta} \]
   fma-def [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\mathsf{fma}\left(\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right), -\sin \phi_1, \cos delta\right)}} \]

3. Applied egg-rr 99.7

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

4. Final simplification 99.7

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

Alternatives

Alternative 1
Error	99.7%
Cost	72064.00

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos delta - \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \cos delta \cdot \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)} \]

Alternative 2
Error	99.7%
Cost	71680.00

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

Alternative 3
Error	94.6%
Cost	65152.00

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta - \sin \phi_1 \cdot \left(t_1 + \cos delta \cdot \sin \phi_1\right)} \end{array} \]

Alternative 4
Error	92.3%
Cost	45504.00

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

Alternative 5
Error	92.0%
Cost	39240.00

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.00013:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\sin theta}{\frac{2}{\frac{t_1}{0.5}}}}{\cos delta}\\ \end{array} \]

Alternative 6
Error	92.0%
Cost	33032.00

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.000295:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\sin theta}{\frac{2}{\frac{t_1}{0.5}}}}{\cos delta}\\ \end{array} \]

Alternative 7
Error	92.0%
Cost	32905.00

\[\begin{array}{l} \mathbf{if}\;delta \leq -2.4 \cdot 10^{-7} \lor \neg \left(delta \leq 0.000196\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 8
Error	88.4%
Cost	32512.00

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

Alternative 9
Error	86.1%
Cost	25984.00

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta} \]

Alternative 10
Error	76.0%
Cost	19849.00

\[\begin{array}{l} \mathbf{if}\;delta \leq -2.75 \cdot 10^{-80} \lor \neg \left(delta \leq 2.8 \cdot 10^{-31}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 11
Error	80.4%
Cost	19849.00

\[\begin{array}{l} \mathbf{if}\;theta \leq -21.5 \lor \neg \left(theta \leq 1.5 \cdot 10^{-12}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 12
Error	68.9%
Cost	19720.00

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.9 \cdot 10^{-295}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 2.4 \cdot 10^{-154}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 13
Error	69.3%
Cost	64.00

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023097 
(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))))))))))