Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.9%

Time: 17.7s

Alternatives: 12

Speedup: 1.3×

Specification

Math

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

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))))))))))

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))))))));
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.sin(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))))

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
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]

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

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))))))))))

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))))))));
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.sin(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))))

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
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]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin theta) (* (cos phi1) (sin delta)))
   (fma
    (pow (cos phi1) 2.0)
    (cos delta)
    (* (* (- (sin phi1)) (* (cos theta) (cos phi1))) (sin delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(theta) * (cos(phi1) * sin(delta))), fma(pow(cos(phi1), 2.0), cos(delta), ((-sin(phi1) * (cos(theta) * cos(phi1))) * sin(delta))));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(cos(phi1) * sin(delta))), fma((cos(phi1) ^ 2.0), cos(delta), Float64(Float64(Float64(-sin(phi1)) * Float64(cos(theta) * cos(phi1))) * sin(delta)))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[((-N[Sin[phi1], $MachinePrecision]) * N[(N[Cos[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $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. Add Preprocessing

4. Applied rewrites 99.6%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \left(\sin delta \cdot \cos \phi_1\right), \cos theta, \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta\right) + \cos delta\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-*.f64 N/A

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 + \phi_1\right), 0.5\right), \cos delta, \sin delta \cdot \left(\left(\cos \phi_1 \cdot \cos theta\right) \cdot \left(-\sin \phi_1\right)\right)\right)} + \lambda_1} \]
9. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \sin delta\\ t_2 := \sin theta \cdot t\_1\\ t_3 := \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \sin^{-1} \left(\cos theta \cdot t\_1 + \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1} + \lambda_1\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 + \phi_1\right), 0.5\right), \cos delta, \left(\left(-\sin \phi_1\right) \cdot \cos \phi_1\right) \cdot \sin delta\right)} + \lambda_1\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, 0.5\right), \cos delta, \left(\left(\sin \phi_1 \cdot \sin delta\right) \cdot \cos \phi_1\right) \cdot \left(-\cos theta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (sin delta)))
        (t_2 (* (sin theta) t_1))
        (t_3
         (+
          (atan2
           (* (* (sin theta) (sin delta)) (cos phi1))
           (-
            (cos delta)
            (*
             (sin (asin (+ (* (cos theta) t_1) (* (sin phi1) (cos delta)))))
             (sin phi1))))
          lambda1)))
   (if (<= t_3 5e-15)
     (+
      (atan2
       t_2
       (fma
        (fma 0.5 (cos (+ phi1 phi1)) 0.5)
        (cos delta)
        (* (* (- (sin phi1)) (cos phi1)) (sin delta))))
      lambda1)
     (if (<= t_3 5.0)
       (atan2
        t_2
        (fma
         (fma (cos (* 2.0 phi1)) 0.5 0.5)
         (cos delta)
         (* (* (* (sin phi1) (sin delta)) (cos phi1)) (- (cos theta)))))
       (+ (atan2 (* theta (sin delta)) (cos delta)) lambda1)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * sin(delta);
	double t_2 = sin(theta) * t_1;
	double t_3 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(asin(((cos(theta) * t_1) + (sin(phi1) * cos(delta))))) * sin(phi1)))) + lambda1;
	double tmp;
	if (t_3 <= 5e-15) {
		tmp = atan2(t_2, fma(fma(0.5, cos((phi1 + phi1)), 0.5), cos(delta), ((-sin(phi1) * cos(phi1)) * sin(delta)))) + lambda1;
	} else if (t_3 <= 5.0) {
		tmp = atan2(t_2, fma(fma(cos((2.0 * phi1)), 0.5, 0.5), cos(delta), (((sin(phi1) * sin(delta)) * cos(phi1)) * -cos(theta))));
	} else {
		tmp = atan2((theta * sin(delta)), cos(delta)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * sin(delta))
	t_2 = Float64(sin(theta) * t_1)
	t_3 = Float64(atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(asin(Float64(Float64(cos(theta) * t_1) + Float64(sin(phi1) * cos(delta))))) * sin(phi1)))) + lambda1)
	tmp = 0.0
	if (t_3 <= 5e-15)
		tmp = Float64(atan(t_2, fma(fma(0.5, cos(Float64(phi1 + phi1)), 0.5), cos(delta), Float64(Float64(Float64(-sin(phi1)) * cos(phi1)) * sin(delta)))) + lambda1);
	elseif (t_3 <= 5.0)
		tmp = atan(t_2, fma(fma(cos(Float64(2.0 * phi1)), 0.5, 0.5), cos(delta), Float64(Float64(Float64(sin(phi1) * sin(delta)) * cos(phi1)) * Float64(-cos(theta)))));
	else
		tmp = Float64(atan(Float64(theta * sin(delta)), cos(delta)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(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[N[ArcSin[N[(N[(N[Cos[theta], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-15], N[(N[ArcTan[t$95$2 / N[(N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[((-N[Sin[phi1], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[ArcTan[t$95$2 / N[(N[(N[Cos[N[(2.0 * phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Cos[theta], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 4.99999999999999999e-15

   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. Add Preprocessing

   4. Applied rewrites 99.7%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \left(\sin delta \cdot \cos \phi_1\right), \cos theta, \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta\right) + \cos delta\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in theta around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-sin.f64 95.9

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

   11. Applied rewrites 95.9%

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

   if 4.99999999999999999e-15 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 5

   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. Add Preprocessing

   4. Applied rewrites 99.6%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \left(\sin delta \cdot \cos \phi_1\right), \cos theta, \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta\right) + \cos delta\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in lambda1 around 0

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

   9. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, 0.5\right), \cos delta, \left(-\cos theta\right) \cdot \left(\left(\sin \phi_1 \cdot \sin delta\right) \cdot \cos \phi_1\right)\right)}} \]

   if 5 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{\cos delta} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in theta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \color{blue}{\sin delta}}{\cos delta} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.0%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{theta}}{\cos delta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1} + \lambda_1 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 + \phi_1\right), 0.5\right), \cos delta, \left(\left(-\sin \phi_1\right) \cdot \cos \phi_1\right) \cdot \sin delta\right)} + \lambda_1\\ \mathbf{elif}\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1} + \lambda_1 \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, 0.5\right), \cos delta, \left(\left(\sin \phi_1 \cdot \sin delta\right) \cdot \cos \phi_1\right) \cdot \left(-\cos theta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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))))))))))