Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 29.1s

Precision: binary64

Cost: 181504

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\\ t_2 := -\sin \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(t_1, t_2, \cos delta\right) + \mathsf{fma}\left(t_2, t_1, \sin \phi_1 \cdot t_1\right)} \end{array} \]

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
 (let* ((t_1
         (fma
          (cos delta)
          (sin phi1)
          (* (sin delta) (* (cos phi1) (cos theta)))))
        (t_2 (- (sin phi1))))
   (+
    lambda1
    (atan2
     (* (sin theta) (* (sin delta) (cos phi1)))
     (+ (fma t_1 t_2 (cos delta)) (fma t_2 t_1 (* (sin phi1) t_1)))))))

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) {
	double t_1 = fma(cos(delta), sin(phi1), (sin(delta) * (cos(phi1) * cos(theta))));
	double t_2 = -sin(phi1);
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (fma(t_1, t_2, cos(delta)) + fma(t_2, t_1, (sin(phi1) * t_1))));
}

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)
	t_1 = fma(cos(delta), sin(phi1), Float64(sin(delta) * Float64(cos(phi1) * cos(theta))))
	t_2 = Float64(-sin(phi1))
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(fma(t_1, t_2, cos(delta)) + fma(t_2, t_1, Float64(sin(phi1) * t_1)))))
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_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = (-N[Sin[phi1], $MachinePrecision])}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$2 + N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.6%

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

2. Simplified 99.6%

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

   [Start] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   associate-*l* [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   cancel-sign-sub-inv [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta + \left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]
   cancel-sign-sub [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta - \left(-\left(-\sin \phi_1\right)\right) \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]
   remove-double-neg [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\sin \phi_1} \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   fma-def [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}} \]
   associate-*l* [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \color{blue}{\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)}\right)\right)} \]

3. Applied egg-rr 99.5%

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

   [Start] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)\right)} \]
   sin-asin [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)}} \]
   add-sqr-sqrt [=>] 52.5%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\left(\sqrt{\sin \phi_1} \cdot \sqrt{\sin \phi_1}\right)} \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)} \]
   sqrt-unprod [=>] 94.7%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\sqrt{\sin \phi_1 \cdot \sin \phi_1}} \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)} \]
   sqr-neg [<=] 94.7%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sqrt{\color{blue}{\left(-\sin \phi_1\right) \cdot \left(-\sin \phi_1\right)}} \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)} \]
   sqrt-unprod [<=] 42.2%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\left(\sqrt{-\sin \phi_1} \cdot \sqrt{-\sin \phi_1}\right)} \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)} \]
   add-sqr-sqrt [<=] 88.1%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\left(-\sin \phi_1\right)} \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)} \]
   fma-udef [=>] 88.1%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(-\sin \phi_1\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos delta + \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)}} \]
   associate-*r* [=>] 88.1%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(-\sin \phi_1\right) \cdot \left(\sin \phi_1 \cdot \cos delta + \color{blue}{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta}\right)} \]
   log1p-expm1-u [=>] 88.1%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin \phi_1\right) \cdot \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \]

4. Applied egg-rr 99.6%

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

   [Start] 99.5%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos delta, \sin \phi_1, \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)} \]
   *-un-lft-identity [=>] 99.5%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{1 \cdot \cos delta} - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos delta, \sin \phi_1, \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)} \]
   log1p-expm1-u [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1 \cdot \cos delta - \color{blue}{\sin \phi_1 \cdot \mathsf{fma}\left(\cos delta, \sin \phi_1, \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}} \]
   fma-udef [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1 \cdot \cos delta - \sin \phi_1 \cdot \color{blue}{\left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}} \]
   *-commutative [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1 \cdot \cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \color{blue}{\left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta}\right)} \]
   associate-*r* [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1 \cdot \cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \color{blue}{\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)}\right)} \]
   +-commutative [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1 \cdot \cos delta - \sin \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right)}} \]
   sin-asin [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{1 \cdot \cos delta - \sin \phi_1 \cdot \color{blue}{\sin \sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right)}} \]
   prod-diff [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\mathsf{fma}\left(1, \cos delta, -\sin \sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) + \mathsf{fma}\left(-\sin \sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right) \cdot \sin \phi_1\right)}} \]

5. Simplified 99.6%

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

   [Start] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right) + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(\cos delta + \left(-\sin \phi_1\right) \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   *-commutative [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos delta + \color{blue}{\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right) \cdot \left(-\sin \phi_1\right)}\right) + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   +-commutative [<=] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right) \cdot \left(-\sin \phi_1\right) + \cos delta\right)} + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   fma-def [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), -\sin \phi_1, \cos delta\right)} + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   fma-udef [=>] 99.7%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(\color{blue}{\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1}, -\sin \phi_1, \cos delta\right) + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   +-commutative [<=] 99.7%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(\color{blue}{\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)}, -\sin \phi_1, \cos delta\right) + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   fma-def [=>] 99.6%	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}, -\sin \phi_1, \cos delta\right) + \mathsf{fma}\left(-\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right), \sin \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]

6. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	181504

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\\ t_2 := -\sin \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(t_1, t_2, \cos delta\right) + \mathsf{fma}\left(t_2, t_1, \sin \phi_1 \cdot t_1\right)} \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	175232

\[\begin{array}{l} t_1 := \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\\ t_2 := \mathsf{fma}\left(\cos delta, \sin \phi_1, t_1\right)\\ t_3 := -\sin \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(t_2, t_3, \cos delta\right) + \mathsf{fma}\left(t_3, t_2, \sin \phi_1 \cdot \left(t_1 + \cos delta \cdot \sin \phi_1\right)\right)} \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	84288

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

Alternative 4
Accuracy	99.8%
Cost	71680

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

Alternative 5
Accuracy	99.8%
Cost	71680

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

Alternative 6
Accuracy	94.8%
Cost	65152

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

Alternative 7
Accuracy	92.6%
Cost	64704

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

Alternative 8
Accuracy	92.6%
Cost	45504

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

Alternative 9
Accuracy	92.3%
Cost	45444

\[\begin{array}{l} t_1 := \sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 185:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]

Alternative 10
Accuracy	92.3%
Cost	39241

\[\begin{array}{l} \mathbf{if}\;delta \leq -6.8 \cdot 10^{-7} \lor \neg \left(delta \leq 160\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \end{array} \]

Alternative 11
Accuracy	89.2%
Cost	32512

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

Alternative 12
Accuracy	87.0%
Cost	25984

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

Alternative 13
Accuracy	80.5%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.105 \lor \neg \left(delta \leq 3.9 \cdot 10^{+82}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]

Alternative 14
Accuracy	75.7%
Cost	19716

\[\begin{array}{l} \mathbf{if}\;theta \leq -66000000000:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 15
Accuracy	70.7%
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023165 
(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))))))))))