Destination given bearing on a great circle

Average Accuracy: 99.7% → 99.7%

Time: 39.4s

Precision: binary64

Cost: 84288

Math

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

↓

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

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
   (* (cos phi1) (* (sin theta) (sin delta)))
   (fma
    (fma (cos phi1) (* (sin delta) (cos theta)) (* (sin phi1) (cos delta)))
    (- (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((cos(phi1) * (sin(theta) * sin(delta))), fma(fma(cos(phi1), (sin(delta) * cos(theta)), (sin(phi1) * cos(delta))), -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(cos(phi1) * Float64(sin(theta) * sin(delta))), fma(fma(cos(phi1), Float64(sin(delta) * cos(theta)), Float64(sin(phi1) * cos(delta))), 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[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $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. Taylor expanded in delta around inf 99.7%

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

3. Simplified 99.7%

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

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

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	77952

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

Alternative 2
Accuracy	99.7%
Cost	71680

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

Alternative 3
Accuracy	94.4%
Cost	71424

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

Alternative 4
Accuracy	94.4%
Cost	71424

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

Alternative 5
Accuracy	94.0%
Cost	65288

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;theta \leq -1.05 \cdot 10^{-36}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta - \log \left(e^{{\sin \phi_1}^{2}}\right)}\\ \mathbf{elif}\;theta \leq 3.5 \cdot 10^{-60}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\left(\cos delta + -0.5\right) + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \end{array} \]

Alternative 6
Accuracy	93.7%
Cost	65288

\[\begin{array}{l} t_1 := \sin \phi_1 \cdot \cos delta\\ \mathbf{if}\;theta \leq -8.5 \cdot 10^{-86}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \left(t_1 + \sin delta \cdot \cos theta\right)}\\ \mathbf{elif}\;theta \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1, t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos delta + -0.5\right) + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \end{array} \]

Alternative 7
Accuracy	94.4%
Cost	65152

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

Alternative 8
Accuracy	94.4%
Cost	65152

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

Alternative 9
Accuracy	93.9%
Cost	59016

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;theta \leq -1.85 \cdot 10^{-36}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta - \log \left(e^{{\sin \phi_1}^{2}}\right)}\\ \mathbf{elif}\;theta \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \sin delta \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\left(\cos delta + -0.5\right) + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \end{array} \]

Alternative 10
Accuracy	91.6%
Cost	58560

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

Alternative 11
Accuracy	91.4%
Cost	39560

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -170000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 0.00033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\frac{1}{\frac{1}{\cos \phi_1 \cdot \cos \phi_1}}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 12
Accuracy	92.1%
Cost	39424

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

Alternative 13
Accuracy	91.4%
Cost	39304

\[\begin{array}{l} \mathbf{if}\;delta \leq -5900000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 0.00033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 14
Accuracy	91.4%
Cost	33160

\[\begin{array}{l} \mathbf{if}\;delta \leq -5900000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 0.00032:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 15
Accuracy	91.8%
Cost	32905

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.0049 \lor \neg \left(delta \leq 0.00039\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 16
Accuracy	91.5%
Cost	32904

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.0062:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 0.00032:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 17
Accuracy	88.4%
Cost	32512

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

Alternative 18
Accuracy	71.3%
Cost	26120

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.55 \cdot 10^{-70}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 9.8 \cdot 10^{-248}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 19
Accuracy	86.4%
Cost	25984

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

Alternative 20
Accuracy	78.6%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;delta \leq -2 \cdot 10^{+89}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 1.46 \cdot 10^{-18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 21
Accuracy	80.6%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -31000000000 \lor \neg \left(delta \leq 0.00053\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \end{array} \]

Alternative 22
Accuracy	67.5%
Cost	19720

\[\begin{array}{l} \mathbf{if}\;theta \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{elif}\;theta \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 23
Accuracy	73.5%
Cost	19584

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

Alternative 24
Accuracy	67.1%
Cost	13184

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

Alternative 25
Accuracy	69.7%
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023137 
(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))))))))))