Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 19.2s

Alternatives: 19

Speedup: N/A×

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: 99.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (fma
    (fma (cos (+ phi1 phi1)) 0.5 0.5)
    (cos delta)
    (* (* (sin delta) (sin phi1)) (* (cos theta) (- (cos phi1))))))
  lambda1))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return atan2((sin(theta) * (sin(delta) * cos(phi1))), fma(fma(cos((phi1 + phi1)), 0.5, 0.5), cos(delta), ((sin(delta) * sin(phi1)) * (cos(theta) * -cos(phi1))))) + lambda1;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), fma(fma(cos(Float64(phi1 + phi1)), 0.5, 0.5), cos(delta), Float64(Float64(sin(delta) * sin(phi1)) * Float64(cos(theta) * Float64(-cos(phi1)))))) + lambda1)
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * (-N[Cos[phi1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $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 egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

   1. distribute-neg-in N/A

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

   2. metadata-eval N/A

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

   3. associate-+l+ N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. +-lowering-+.f64 99.8

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin \phi_1\\ t_2 := \mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), -0.5, 0.5\right)\\ t_3 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ t_4 := \sin delta \cdot \cos \phi_1\\ t_5 := \lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + t\_4 \cdot \cos theta\right)}\\ \mathbf{if}\;t\_5 \leq 0.05:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\cos \phi_1 \cdot t\_1, \sin delta, \cos delta - \cos delta \cdot t\_2\right)}\\ \mathbf{elif}\;t\_5 \leq 3.12:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right), \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t\_4}{\mathsf{fma}\left(1 - t\_2, \cos delta, \left(\sin delta \cdot \sin \phi_1\right) \cdot \left(-\cos \phi_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- (sin phi1)))
        (t_2 (fma (cos (+ phi1 phi1)) -0.5 0.5))
        (t_3 (* (cos phi1) (* (sin theta) (sin delta))))
        (t_4 (* (sin delta) (cos phi1)))
        (t_5
         (+
          lambda1
          (atan2
           t_3
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin (+ (* (cos delta) (sin phi1)) (* t_4 (cos theta)))))))))))
   (if (<= t_5 0.05)
     (+
      lambda1
      (atan2
       t_3
       (fma
        (* (cos phi1) t_1)
        (sin delta)
        (- (cos delta) (* (cos delta) t_2)))))
     (if (<= t_5 3.12)
       (atan2
        t_3
        (fma
         (fma 0.5 (cos (* phi1 -2.0)) 0.5)
         (cos delta)
         (* (cos phi1) (* (sin delta) (* (cos theta) t_1)))))
       (+
        lambda1
        (atan2
         (* (sin theta) t_4)
         (fma
          (- 1.0 t_2)
          (cos delta)
          (* (* (sin delta) (sin phi1)) (- (cos phi1))))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = -sin(phi1);
	double t_2 = fma(cos((phi1 + phi1)), -0.5, 0.5);
	double t_3 = cos(phi1) * (sin(theta) * sin(delta));
	double t_4 = sin(delta) * cos(phi1);
	double t_5 = lambda1 + atan2(t_3, (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (t_4 * cos(theta))))))));
	double tmp;
	if (t_5 <= 0.05) {
		tmp = lambda1 + atan2(t_3, fma((cos(phi1) * t_1), sin(delta), (cos(delta) - (cos(delta) * t_2))));
	} else if (t_5 <= 3.12) {
		tmp = atan2(t_3, fma(fma(0.5, cos((phi1 * -2.0)), 0.5), cos(delta), (cos(phi1) * (sin(delta) * (cos(theta) * t_1)))));
	} else {
		tmp = lambda1 + atan2((sin(theta) * t_4), fma((1.0 - t_2), cos(delta), ((sin(delta) * sin(phi1)) * -cos(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(-sin(phi1))
	t_2 = fma(cos(Float64(phi1 + phi1)), -0.5, 0.5)
	t_3 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	t_4 = Float64(sin(delta) * cos(phi1))
	t_5 = Float64(lambda1 + atan(t_3, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(t_4 * cos(theta)))))))))
	tmp = 0.0
	if (t_5 <= 0.05)
		tmp = Float64(lambda1 + atan(t_3, fma(Float64(cos(phi1) * t_1), sin(delta), Float64(cos(delta) - Float64(cos(delta) * t_2)))));
	elseif (t_5 <= 3.12)
		tmp = atan(t_3, fma(fma(0.5, cos(Float64(phi1 * -2.0)), 0.5), cos(delta), Float64(cos(phi1) * Float64(sin(delta) * Float64(cos(theta) * t_1)))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * t_4), fma(Float64(1.0 - t_2), cos(delta), Float64(Float64(sin(delta) * sin(phi1)) * Float64(-cos(phi1))))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = (-N[Sin[phi1], $MachinePrecision])}, Block[{t$95$2 = N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.05], N[(lambda1 + N[ArcTan[t$95$3 / N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] - N[(N[Cos[delta], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.12], N[ArcTan[t$95$3 / N[(N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$4), $MachinePrecision] / N[(N[(1.0 - t$95$2), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Cos[phi1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))))))))) < 0.050000000000000003

   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 egg-rr 99.8%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. cos-2 N/A

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

      12. cos-sum N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. cos-lowering-cos.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in theta around 0

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

      1. cos-lowering-cos.f64 95.9

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

   9. Simplified 95.9%

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

   if 0.050000000000000003 < (+.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))))))))) < 3.1200000000000001

   1. Initial program 98.9%

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.3%

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

   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. Simplified 96.7%

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

   if 3.1200000000000001 < (+.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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in theta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. sin-lowering-sin.f64 99.7

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

   9. Simplified 99.7%

      \[\leadsto \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), -0.5, 0.5\right)\right) + 1, \cos delta, \color{blue}{\cos \phi_1 \cdot \left(\sin delta \cdot \left(-\sin \phi_1\right)\right)}\right)} + \lambda_1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)} \leq 0.05:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\mathsf{fma}\left(\cos \phi_1 \cdot \left(-\sin \phi_1\right), \sin delta, \cos delta - \cos delta \cdot \mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), -0.5, 0.5\right)\right)}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)} \leq 3.12:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right), \cos delta, \cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot \left(-\sin \phi_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(1 - \mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), -0.5, 0.5\right), \cos delta, \left(\sin delta \cdot \sin \phi_1\right) \cdot \left(-\cos \phi_1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)}\\ \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta)))
        (t_2
         (+
          lambda1
          (atan2
           (* (cos phi1) t_1)
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin
               (+
                (* (cos delta) (sin phi1))
                (* (* (sin delta) (cos phi1)) (cos theta)))))))))))
   (if (<= t_2 -50.0)
     lambda1
     (if (<= t_2 -2e-14)
       (atan2 t_1 (cos delta))
       (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double t_2 = lambda1 + atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + ((sin(delta) * cos(phi1)) * cos(theta))))))));
	double tmp;
	if (t_2 <= -50.0) {
		tmp = lambda1;
	} else if (t_2 <= -2e-14) {
		tmp = atan2(t_1, cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(theta) * sin(delta)
    t_2 = lambda1 + atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + ((sin(delta) * cos(phi1)) * cos(theta))))))))
    if (t_2 <= (-50.0d0)) then
        tmp = lambda1
    else if (t_2 <= (-2d-14)) then
        tmp = atan2(t_1, cos(delta))
    else
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.sin(theta) * Math.sin(delta);
	double t_2 = lambda1 + Math.atan2((Math.cos(phi1) * t_1), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.cos(delta) * Math.sin(phi1)) + ((Math.sin(delta) * Math.cos(phi1)) * Math.cos(theta))))))));
	double tmp;
	if (t_2 <= -50.0) {
		tmp = lambda1;
	} else if (t_2 <= -2e-14) {
		tmp = Math.atan2(t_1, Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(theta) * math.sin(delta)
	t_2 = lambda1 + math.atan2((math.cos(phi1) * t_1), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.cos(delta) * math.sin(phi1)) + ((math.sin(delta) * math.cos(phi1)) * math.cos(theta))))))))
	tmp = 0
	if t_2 <= -50.0:
		tmp = lambda1
	elif t_2 <= -2e-14:
		tmp = math.atan2(t_1, math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	t_2 = Float64(lambda1 + atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(Float64(sin(delta) * cos(phi1)) * cos(theta)))))))))
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = lambda1;
	elseif (t_2 <= -2e-14)
		tmp = atan(t_1, cos(delta));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(theta) * sin(delta);
	t_2 = lambda1 + atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + ((sin(delta) * cos(phi1)) * cos(theta))))))));
	tmp = 0.0;
	if (t_2 <= -50.0)
		tmp = lambda1;
	elseif (t_2 <= -2e-14)
		tmp = atan2(t_1, cos(delta));
	else
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -50.0], lambda1, If[LessEqual[t$95$2, -2e-14], N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $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))))))))) < -50

   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 lambda1 around inf

      \[\leadsto \color{blue}{\lambda_1} \]
   4. Step-by-step derivation

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\lambda_1} \]

   if -50 < (+.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))))))))) < -2e-14

   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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 68.0

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

   8. Simplified 68.0%

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

   9. Taylor expanded in lambda1 around 0

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

       1. atan2-lowering-atan2.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sin-lowering-sin.f64 N/A

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

       4. sin-lowering-sin.f64 N/A

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

       5. cos-lowering-cos.f64 64.4

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

   11. Simplified 64.4%

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

   if -2e-14 < (+.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 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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 79.2

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

   8. Simplified 79.2%

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

   9. Taylor expanded in delta around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sin-lowering-sin.f64 71.5

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

   11. Simplified 71.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)} \leq -50:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (fma
    (* (cos phi1) (- (sin phi1)))
    (sin delta)
    (- (cos delta) (* (cos delta) (fma (cos (+ phi1 phi1)) -0.5 0.5)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), fma((cos(phi1) * -sin(phi1)), sin(delta), (cos(delta) - (cos(delta) * fma(cos((phi1 + phi1)), -0.5, 0.5)))));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), fma(Float64(cos(phi1) * Float64(-sin(phi1))), sin(delta), Float64(cos(delta) - Float64(cos(delta) * fma(cos(Float64(phi1 + phi1)), -0.5, 0.5))))))
end

Wolfram

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[Sin[phi1], $MachinePrecision])), $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] - N[(N[Cos[delta], $MachinePrecision] * N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $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 egg-rr 99.8%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. +-lowering-+.f64 N/A

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

   15. metadata-eval N/A

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

   16. cos-lowering-cos.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

   1. cos-lowering-cos.f64 93.4

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

9. Simplified 93.4%

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

10. Final simplification 93.4%

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

Alternative 5: 94.3% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (fma
    (- 1.0 (fma (cos (+ phi1 phi1)) -0.5 0.5))
    (cos delta)
    (* (* (sin delta) (sin phi1)) (- (cos phi1)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), fma((1.0 - fma(cos((phi1 + phi1)), -0.5, 0.5)), cos(delta), ((sin(delta) * sin(phi1)) * -cos(phi1))));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), fma(Float64(1.0 - fma(cos(Float64(phi1 + phi1)), -0.5, 0.5)), cos(delta), Float64(Float64(sin(delta) * sin(phi1)) * Float64(-cos(phi1))))))
end

Wolfram

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[(1.0 - N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Cos[phi1], $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 egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. mul-1-neg N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. sin-lowering-sin.f64 93.4

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

9. Simplified 93.4%

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

10. Final simplification 93.4%

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

Alternative 6: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{-0.5 + \mathsf{fma}\left(\sin delta \cdot \sin \phi_1, -\cos \phi_1, \mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), \cos delta\right)\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (+
    -0.5
    (fma
     (* (sin delta) (sin phi1))
     (- (cos phi1))
     (fma 0.5 (cos (* phi1 -2.0)) (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (-0.5 + fma((sin(delta) * sin(phi1)), -cos(phi1), fma(0.5, cos((phi1 * -2.0)), cos(delta)))));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(-0.5 + fma(Float64(sin(delta) * sin(phi1)), Float64(-cos(phi1)), fma(0.5, cos(Float64(phi1 * -2.0)), cos(delta))))))
end

Wolfram

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[(-0.5 + N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Cos[phi1], $MachinePrecision]) + N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[delta], $MachinePrecision]), $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 egg-rr 99.8%

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

5. Taylor expanded in delta around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 91.5

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

7. Simplified 91.5%

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

8. Taylor expanded in theta around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. mul-1-neg N/A

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

   11. +-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

10. Simplified 91.2%

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

11. Final simplification 91.2%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{-0.5 + \mathsf{fma}\left(\sin delta \cdot \sin \phi_1, -\cos \phi_1, \mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), \cos delta\right)\right)} \]
12. Add Preprocessing

Alternative 7: 92.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) ** 2.0d0)))
end function

Java

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

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), (math.cos(delta) - math.pow(math.sin(phi1), 2.0)))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) ^ 2.0)));
end

Wolfram

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[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $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

3. Taylor expanded in delta around 0

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

   1. pow-lowering-pow.f64 N/A

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

   2. sin-lowering-sin.f64 90.6

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

5. Simplified 90.6%

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

6. Final simplification 90.6%

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

Alternative 8: 91.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))))
   (if (<= delta -4.1e+23)
     t_1
     (if (<= delta 1.25e-27)
       (+
        lambda1
        (atan2
         (* (sin theta) (* (sin delta) (cos phi1)))
         (fma 0.5 (cos (* phi1 -2.0)) 0.5)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	double tmp;
	if (delta <= -4.1e+23) {
		tmp = t_1;
	} else if (delta <= 1.25e-27) {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), fma(0.5, cos((phi1 * -2.0)), 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)))
	tmp = 0.0
	if (delta <= -4.1e+23)
		tmp = t_1;
	elseif (delta <= 1.25e-27)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), fma(0.5, cos(Float64(phi1 * -2.0)), 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -4.1e+23], t$95$1, If[LessEqual[delta, 1.25e-27], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.09999999999999996e23 or 1.25e-27 < delta

   1. Initial program 99.8%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   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. cos-lowering-cos.f64 83.5

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

   5. Simplified 83.5%

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

   if -4.09999999999999996e23 < delta < 1.25e-27

   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 egg-rr 99.7%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 97.9

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

   10. Simplified 97.9%

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

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

       3. associate-*r* N/A

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

       4. atan2-lowering-atan2.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sin-lowering-sin.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. sin-lowering-sin.f64 N/A

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

       9. cos-lowering-cos.f64 N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. cos-lowering-cos.f64 N/A

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

       12. *-lowering-*.f64 97.9

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

   12. Applied egg-rr 97.9%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)} + \lambda_1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -4.1e+23)
     t_2
     (if (<= delta 1.25e-27)
       (+ lambda1 (atan2 t_1 (fma 0.5 (cos (* phi1 -2.0)) 0.5)))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -4.1e+23) {
		tmp = t_2;
	} else if (delta <= 1.25e-27) {
		tmp = lambda1 + atan2(t_1, fma(0.5, cos((phi1 * -2.0)), 0.5));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -4.1e+23)
		tmp = t_2;
	elseif (delta <= 1.25e-27)
		tmp = Float64(lambda1 + atan(t_1, fma(0.5, cos(Float64(phi1 * -2.0)), 0.5)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -4.1e+23], t$95$2, If[LessEqual[delta, 1.25e-27], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.09999999999999996e23 or 1.25e-27 < delta

   1. Initial program 99.8%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   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. cos-lowering-cos.f64 83.5

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

   5. Simplified 83.5%

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

   if -4.09999999999999996e23 < delta < 1.25e-27

   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 egg-rr 99.7%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 97.9

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

   10. Simplified 97.9%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))))
   (if (<= delta -4.1e+23)
     t_1
     (if (<= delta 1.25e-27)
       (+
        lambda1
        (atan2
         (* (sin delta) (* (sin theta) (cos phi1)))
         (fma 0.5 (cos (* phi1 -2.0)) 0.5)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	double tmp;
	if (delta <= -4.1e+23) {
		tmp = t_1;
	} else if (delta <= 1.25e-27) {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), fma(0.5, cos((phi1 * -2.0)), 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)))
	tmp = 0.0
	if (delta <= -4.1e+23)
		tmp = t_1;
	elseif (delta <= 1.25e-27)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), fma(0.5, cos(Float64(phi1 * -2.0)), 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -4.1e+23], t$95$1, If[LessEqual[delta, 1.25e-27], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.09999999999999996e23 or 1.25e-27 < delta

   1. Initial program 99.8%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   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. cos-lowering-cos.f64 83.5

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

   5. Simplified 83.5%

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

   if -4.09999999999999996e23 < delta < 1.25e-27

   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 egg-rr 99.7%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 97.9

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

   10. Simplified 97.9%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sin-lowering-sin.f64 N/A

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

       7. cos-lowering-cos.f64 N/A

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

       8. sin-lowering-sin.f64 97.9

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

   12. Applied egg-rr 97.9%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\sin theta \cdot \cos \phi_1\right) \cdot \sin delta}}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -0.000215:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))))
   (if (<= delta -0.000215)
     t_1
     (if (<= delta 5e-29)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin theta) delta))
         (fma 0.5 (cos (* phi1 -2.0)) 0.5)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	double tmp;
	if (delta <= -0.000215) {
		tmp = t_1;
	} else if (delta <= 5e-29) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), fma(0.5, cos((phi1 * -2.0)), 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)))
	tmp = 0.0
	if (delta <= -0.000215)
		tmp = t_1;
	elseif (delta <= 5e-29)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), fma(0.5, cos(Float64(phi1 * -2.0)), 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.000215], t$95$1, If[LessEqual[delta, 5e-29], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.14999999999999995e-4 or 4.99999999999999986e-29 < delta

   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

   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. cos-lowering-cos.f64 82.4

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

   5. Simplified 82.4%

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

   if -2.14999999999999995e-4 < delta < 4.99999999999999986e-29

   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 egg-rr 99.8%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 99.8

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

   7. Simplified 99.8%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.8

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

   10. Simplified 99.8%

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

   11. Taylor expanded in delta around 0

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

   13. Simplified 99.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \color{blue}{delta}\right) \cdot \cos \phi_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.000215:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin theta) (sin delta)) (cos delta)))))
   (if (<= delta -4.1e+23)
     t_1
     (if (<= delta 1.25e-27)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin theta) delta))
         (fma 0.5 (cos (* phi1 -2.0)) 0.5)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -4.1e+23) {
		tmp = t_1;
	} else if (delta <= 1.25e-27) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), fma(0.5, cos((phi1 * -2.0)), 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -4.1e+23)
		tmp = t_1;
	elseif (delta <= 1.25e-27)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), fma(0.5, cos(Float64(phi1 * -2.0)), 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -4.1e+23], t$95$1, If[LessEqual[delta, 1.25e-27], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.09999999999999996e23 or 1.25e-27 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 82.2

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

   5. Simplified 82.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 80.1

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

   8. Simplified 80.1%

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

   if -4.09999999999999996e23 < delta < 1.25e-27

   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 egg-rr 99.7%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 97.9

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

   10. Simplified 97.9%

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

   11. Taylor expanded in delta around 0

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

   13. Simplified 97.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \color{blue}{delta}\right) \cdot \cos \phi_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.1 \cdot 10^{+23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq 1350000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 1350000000000.0)
   (+ lambda1 (atan2 (* (sin theta) (sin delta)) (cos delta)))
   (+
    lambda1
    (atan2
     (* (cos phi1) (* theta (sin delta)))
     (fma 0.5 (cos (* phi1 -2.0)) 0.5)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= 1350000000000.0) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (theta * sin(delta))), fma(0.5, cos((phi1 * -2.0)), 0.5));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= 1350000000000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(theta * sin(delta))), fma(0.5, cos(Float64(phi1 * -2.0)), 0.5)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, 1350000000000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 1.35e12

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 91.4

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

   5. Simplified 91.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 91.1

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

   8. Simplified 91.1%

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

   if 1.35e12 < phi1

   1. Initial program 99.3%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in delta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 82.9

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

   7. Simplified 82.9%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 76.7

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

   10. Simplified 76.7%

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

   11. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sin-lowering-sin.f64 68.2

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

   13. Simplified 68.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\sin delta \cdot theta\right)} \cdot \cos \phi_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 1350000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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(delta))
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(delta));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $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

3. Taylor expanded in phi1 around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 N/A

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

   3. sin-lowering-sin.f64 84.4

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

5. Simplified 84.4%

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

6. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 83.4

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

8. Simplified 83.4%

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

9. Final simplification 83.4%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta} \]
10. Add Preprocessing

Alternative 15: 80.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -9500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.5 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(theta \cdot theta, 0.008333333333333333, -0.16666666666666666\right), theta \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -9500000.0)
   (+
    lambda1
    (atan2
     (* (sin delta) (fma theta (* -0.16666666666666666 (* theta theta)) theta))
     (cos delta)))
   (if (<= delta 4.5e-22)
     (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
     (+
      lambda1
      (atan2
       (*
        (sin delta)
        (fma
         (fma (* theta theta) 0.008333333333333333 -0.16666666666666666)
         (* theta (* theta theta))
         theta))
       (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -9500000.0) {
		tmp = lambda1 + atan2((sin(delta) * fma(theta, (-0.16666666666666666 * (theta * theta)), theta)), cos(delta));
	} else if (delta <= 4.5e-22) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * fma(fma((theta * theta), 0.008333333333333333, -0.16666666666666666), (theta * (theta * theta)), theta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -9500000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * fma(theta, Float64(-0.16666666666666666 * Float64(theta * theta)), theta)), cos(delta)));
	elseif (delta <= 4.5e-22)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * fma(fma(Float64(theta * theta), 0.008333333333333333, -0.16666666666666666), Float64(theta * Float64(theta * theta)), theta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -9500000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 4.5e-22], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[(N[(theta * theta), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(theta * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -9.5e6

   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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 76.3

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

   5. Simplified 76.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 73.9

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

   8. Simplified 73.9%

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

   9. Taylor expanded in theta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\left(theta \cdot \left(1 + \frac{-1}{6} \cdot {theta}^{2}\right)\right)}}{\cos delta} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \color{blue}{\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right)}\right)}{\cos delta} \]

       2. distribute-lft-in N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + theta \cdot 1\right)}}{\cos delta} \]

       3. *-rgt-identity N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + \color{blue}{theta}\right)}{\cos delta} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(theta, \frac{-1}{6} \cdot {theta}^{2}, theta\right)}}{\cos delta} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, \color{blue}{\frac{-1}{6} \cdot {theta}^{2}}, theta\right)}{\cos delta} \]

       6. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, \frac{-1}{6} \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

       7. *-lowering-*.f64 61.0

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

   11. Simplified 61.0%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}}{\cos delta} \]

   if -9.5e6 < delta < 4.49999999999999987e-22

   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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 87.5

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

   8. Simplified 87.5%

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

   9. Taylor expanded in delta around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sin-lowering-sin.f64 87.4

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

   11. Simplified 87.4%

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

   if 4.49999999999999987e-22 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 84.7

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

   5. Simplified 84.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 82.9

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

   8. Simplified 82.9%

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

   9. Taylor expanded in theta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\left(theta \cdot \left(1 + {theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right)\right)\right)}}{\cos delta} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \color{blue}{\left({theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) + 1\right)}\right)}{\cos delta} \]

       2. distribute-rgt-in N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\left(\left({theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right)\right) \cdot theta + 1 \cdot theta\right)}}{\cos delta} \]

       3. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\color{blue}{\left(\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot {theta}^{2}\right)} \cdot theta + 1 \cdot theta\right)}{\cos delta} \]

       4. associate-*l* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot \left({theta}^{2} \cdot theta\right)} + 1 \cdot theta\right)}{\cos delta} \]

       5. pow-plus N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{theta}^{\left(2 + 1\right)}} + 1 \cdot theta\right)}{\cos delta} \]

       6. metadata-eval N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot {theta}^{\color{blue}{3}} + 1 \cdot theta\right)}{\cos delta} \]

       7. cube-unmult N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(theta \cdot \left(theta \cdot theta\right)\right)} + 1 \cdot theta\right)}{\cos delta} \]

       8. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot \left(theta \cdot \color{blue}{{theta}^{2}}\right) + 1 \cdot theta\right)}{\cos delta} \]

       9. *-lft-identity N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}\right) \cdot \left(theta \cdot {theta}^{2}\right) + \color{blue}{theta}\right)}{\cos delta} \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {theta}^{2} - \frac{1}{6}, theta \cdot {theta}^{2}, theta\right)}}{\cos delta} \]

       11. sub-neg N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {theta}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, theta \cdot {theta}^{2}, theta\right)}{\cos delta} \]

       12. *-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\color{blue}{{theta}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), theta \cdot {theta}^{2}, theta\right)}{\cos delta} \]

       13. metadata-eval N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left({theta}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, theta \cdot {theta}^{2}, theta\right)}{\cos delta} \]

       14. accelerator-lowering-fma.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({theta}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, theta \cdot {theta}^{2}, theta\right)}{\cos delta} \]

       15. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{theta \cdot theta}, \frac{1}{120}, \frac{-1}{6}\right), theta \cdot {theta}^{2}, theta\right)}{\cos delta} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{theta \cdot theta}, \frac{1}{120}, \frac{-1}{6}\right), theta \cdot {theta}^{2}, theta\right)}{\cos delta} \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(theta \cdot theta, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{theta \cdot {theta}^{2}}, theta\right)}{\cos delta} \]

       18. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(theta \cdot theta, \frac{1}{120}, \frac{-1}{6}\right), theta \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

       19. *-lowering-*.f64 71.8

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(theta \cdot theta, 0.008333333333333333, -0.16666666666666666\right), theta \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

   11. Simplified 71.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(theta \cdot theta, 0.008333333333333333, -0.16666666666666666\right), theta \cdot \left(theta \cdot theta\right), theta\right)}}{\cos delta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -9500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.5 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(\mathsf{fma}\left(theta \cdot theta, 0.008333333333333333, -0.16666666666666666\right), theta \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -15000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.5 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -15000000.0)
   (+
    lambda1
    (atan2
     (* (sin delta) (fma theta (* -0.16666666666666666 (* theta theta)) theta))
     (cos delta)))
   (if (<= delta 4.5e-22)
     (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
     (+ lambda1 (atan2 (* theta (sin delta)) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -15000000.0) {
		tmp = lambda1 + atan2((sin(delta) * fma(theta, (-0.16666666666666666 * (theta * theta)), theta)), cos(delta));
	} else if (delta <= 4.5e-22) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -15000000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * fma(theta, Float64(-0.16666666666666666 * Float64(theta * theta)), theta)), cos(delta)));
	elseif (delta <= 4.5e-22)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -15000000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 4.5e-22], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.5e7

   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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 76.3

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

   5. Simplified 76.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 73.9

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

   8. Simplified 73.9%

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

   9. Taylor expanded in theta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\left(theta \cdot \left(1 + \frac{-1}{6} \cdot {theta}^{2}\right)\right)}}{\cos delta} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \color{blue}{\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right)}\right)}{\cos delta} \]

       2. distribute-lft-in N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + theta \cdot 1\right)}}{\cos delta} \]

       3. *-rgt-identity N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + \color{blue}{theta}\right)}{\cos delta} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(theta, \frac{-1}{6} \cdot {theta}^{2}, theta\right)}}{\cos delta} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, \color{blue}{\frac{-1}{6} \cdot {theta}^{2}}, theta\right)}{\cos delta} \]

       6. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, \frac{-1}{6} \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

       7. *-lowering-*.f64 61.0

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

   11. Simplified 61.0%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}}{\cos delta} \]

   if -1.5e7 < delta < 4.49999999999999987e-22

   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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 87.5

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

   8. Simplified 87.5%

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

   9. Taylor expanded in delta around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sin-lowering-sin.f64 87.4

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

   11. Simplified 87.4%

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

   if 4.49999999999999987e-22 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 84.7

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

   5. Simplified 84.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 82.9

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

   8. Simplified 82.9%

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

   9. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sin-lowering-sin.f64 71.4

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

   11. Simplified 71.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -15000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.5 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 77.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 4.5 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 4.5e-22)
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 4.5e-22) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (delta <= 4.5d-22) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 4.5e-22) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= 4.5e-22:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 4.5e-22)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= 4.5e-22)
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 4.5e-22], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < 4.49999999999999987e-22

   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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 83.5

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

   8. Simplified 83.5%

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

   9. Taylor expanded in delta around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sin-lowering-sin.f64 77.3

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

   11. Simplified 77.3%

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

   if 4.49999999999999987e-22 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. sin-lowering-sin.f64 84.7

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

   5. Simplified 84.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 82.9

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

   8. Simplified 82.9%

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

   9. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sin-lowering-sin.f64 71.4

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

   11. Simplified 71.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot theta}}{\cos delta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq 4.5 \cdot 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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) * delta), cos(delta))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $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

3. Taylor expanded in phi1 around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 N/A

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

   3. sin-lowering-sin.f64 84.4

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

5. Simplified 84.4%

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

6. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 83.4

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

8. Simplified 83.4%

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

9. Taylor expanded in delta around 0

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

    1. *-lowering-*.f64 N/A

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

    2. sin-lowering-sin.f64 72.9

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

11. Simplified 72.9%

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

12. Final simplification 72.9%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta} \]
13. Add Preprocessing

Alternative 19: 70.5% accurate, 1341.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta) :precision binary64 lambda1)

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

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
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return lambda1
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := lambda1

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

3. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{\lambda_1} \]
4. Step-by-step derivation

5. Simplified 67.6%

   \[\leadsto \color{blue}{\lambda_1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))))))))