Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.7%

Time: 21.3s

Alternatives: 15

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: 99.7% 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: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot \sin theta}{\mathsf{fma}\left(t\_1 \cdot \left(0 - \sin \phi_1\right), \cos theta, \cos delta - \cos delta \cdot \left(0.5 + -0.5 \cdot \cos \left(\phi_1 \cdot 2\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (+
    lambda1
    (atan2
     (* t_1 (sin theta))
     (fma
      (* t_1 (- 0.0 (sin phi1)))
      (cos theta)
      (- (cos delta) (* (cos delta) (+ 0.5 (* -0.5 (cos (* phi1 2.0)))))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * cos(phi1);
	return lambda1 + atan2((t_1 * sin(theta)), fma((t_1 * (0.0 - sin(phi1))), cos(theta), (cos(delta) - (cos(delta) * (0.5 + (-0.5 * cos((phi1 * 2.0))))))));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	return Float64(lambda1 + atan(Float64(t_1 * sin(theta)), fma(Float64(t_1 * Float64(0.0 - sin(phi1))), cos(theta), Float64(cos(delta) - Float64(cos(delta) * Float64(0.5 + Float64(-0.5 * cos(Float64(phi1 * 2.0)))))))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(t$95$1 * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(0.0 - N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] - N[(N[Cos[delta], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\left(\cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\left(\mathsf{neg}\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right) + \color{blue}{\left(\cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\left(\mathsf{neg}\left(\sin delta \cdot \cos \phi_1\right)\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \left(\color{blue}{\cos delta} - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\left(\mathsf{neg}\left(\sin delta \cdot \cos \phi_1\right)\right) \cdot \left(\sin \phi_1 \cdot \cos theta\right) + \left(\cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\left(\left(\mathsf{neg}\left(\sin delta \cdot \cos \phi_1\right)\right) \cdot \sin \phi_1\right) \cdot \cos theta + \left(\color{blue}{\cos delta} - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right)\right)\right)\right) \]

   6. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\mathsf{fma}\left(\left(\mathsf{neg}\left(\sin delta \cdot \cos \phi_1\right)\right) \cdot \sin \phi_1, \color{blue}{\cos theta}, \cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\sin delta \cdot \cos \phi_1\right)\right) \cdot \sin \phi_1\right), \color{blue}{\cos theta}, \left(\cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta}{\mathsf{fma}\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(0 - \sin \phi_1\right), \cos theta, \cos delta - \cos delta \cdot \left(0.5 + -0.5 \cdot \cos \left(\phi_1 \cdot 2\right)\right)\right)} \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot \sin theta}{\left(\cos delta + \cos delta \cdot \left(0.5 \cdot \cos \left(\phi_1 \cdot 2\right) - 0.5\right)\right) - t\_1 \cdot \left(\sin \phi_1 \cdot \cos theta\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (+
    lambda1
    (atan2
     (* t_1 (sin theta))
     (-
      (+ (cos delta) (* (cos delta) (- (* 0.5 (cos (* phi1 2.0))) 0.5)))
      (* t_1 (* (sin phi1) (cos theta))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * cos(phi1);
	return lambda1 + atan2((t_1 * sin(theta)), ((cos(delta) + (cos(delta) * ((0.5 * cos((phi1 * 2.0))) - 0.5))) - (t_1 * (sin(phi1) * 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
    real(8) :: t_1
    t_1 = sin(delta) * cos(phi1)
    code = lambda1 + atan2((t_1 * sin(theta)), ((cos(delta) + (cos(delta) * ((0.5d0 * cos((phi1 * 2.0d0))) - 0.5d0))) - (t_1 * (sin(phi1) * cos(theta)))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.sin(delta) * Math.cos(phi1);
	return lambda1 + Math.atan2((t_1 * Math.sin(theta)), ((Math.cos(delta) + (Math.cos(delta) * ((0.5 * Math.cos((phi1 * 2.0))) - 0.5))) - (t_1 * (Math.sin(phi1) * Math.cos(theta)))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * math.cos(phi1)
	return lambda1 + math.atan2((t_1 * math.sin(theta)), ((math.cos(delta) + (math.cos(delta) * ((0.5 * math.cos((phi1 * 2.0))) - 0.5))) - (t_1 * (math.sin(phi1) * math.cos(theta)))))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	return Float64(lambda1 + atan(Float64(t_1 * sin(theta)), Float64(Float64(cos(delta) + Float64(cos(delta) * Float64(Float64(0.5 * cos(Float64(phi1 * 2.0))) - 0.5))) - Float64(t_1 * Float64(sin(phi1) * cos(theta))))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * cos(phi1);
	tmp = lambda1 + atan2((t_1 * sin(theta)), ((cos(delta) + (cos(delta) * ((0.5 * cos((phi1 * 2.0))) - 0.5))) - (t_1 * (sin(phi1) * cos(theta)))));
end

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), ((cos(delta) + (cos(delta) * ((0.5 * cos((phi1 * 2.0))) - 0.5))) - ((sin(delta) * cos(phi1)) * (sin(phi1) * 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(delta) * (cos(phi1) * sin(theta))), ((cos(delta) + (cos(delta) * ((0.5d0 * cos((phi1 * 2.0d0))) - 0.5d0))) - ((sin(delta) * cos(phi1)) * (sin(phi1) * 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(delta) * (Math.cos(phi1) * Math.sin(theta))), ((Math.cos(delta) + (Math.cos(delta) * ((0.5 * Math.cos((phi1 * 2.0))) - 0.5))) - ((Math.sin(delta) * Math.cos(phi1)) * (Math.sin(phi1) * Math.cos(theta)))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(delta) * (math.cos(phi1) * math.sin(theta))), ((math.cos(delta) + (math.cos(delta) * ((0.5 * math.cos((phi1 * 2.0))) - 0.5))) - ((math.sin(delta) * math.cos(phi1)) * (math.sin(phi1) * math.cos(theta)))))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), ((cos(delta) + (cos(delta) * ((0.5 * cos((phi1 * 2.0))) - 0.5))) - ((sin(delta) * cos(phi1)) * (sin(phi1) * cos(theta)))));
end

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\cos \phi_1 \cdot \sin theta\right), \sin delta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \sin theta\right), \sin delta\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \sin theta\right), \sin delta\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(theta\right)\right), \sin delta\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(theta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

   1. sin-asin N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \color{blue}{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta}\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \color{blue}{\left(\cos \phi_1 \cdot \sin delta\right)} \cdot \cos theta\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \cos theta\right)}\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \color{blue}{\sin \phi_1}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right), \color{blue}{\sin \phi_1}\right)\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 94.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin delta) (cos phi1)) (sin theta))
   (-
    (* (cos delta) (- 1.0 (+ 0.5 (* -0.5 (cos (* phi1 2.0))))))
    (* (cos phi1) (* (sin delta) (sin phi1)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(delta) * cos(phi1)) * sin(theta)), ((cos(delta) * (1.0 - (0.5 + (-0.5 * cos((phi1 * 2.0)))))) - (cos(phi1) * (sin(delta) * sin(phi1)))));
}

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(delta) * cos(phi1)) * sin(theta)), ((cos(delta) * (1.0d0 - (0.5d0 + ((-0.5d0) * cos((phi1 * 2.0d0)))))) - (cos(phi1) * (sin(delta) * sin(phi1)))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(((Math.sin(delta) * Math.cos(phi1)) * Math.sin(theta)), ((Math.cos(delta) * (1.0 - (0.5 + (-0.5 * Math.cos((phi1 * 2.0)))))) - (Math.cos(phi1) * (Math.sin(delta) * Math.sin(phi1)))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(delta) * math.cos(phi1)) * math.sin(theta)), ((math.cos(delta) * (1.0 - (0.5 + (-0.5 * math.cos((phi1 * 2.0)))))) - (math.cos(phi1) * (math.sin(delta) * math.sin(phi1)))))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(delta) * cos(phi1)) * sin(theta)), Float64(Float64(cos(delta) * Float64(1.0 - Float64(0.5 + Float64(-0.5 * cos(Float64(phi1 * 2.0)))))) - Float64(cos(phi1) * Float64(sin(delta) * sin(phi1))))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(delta) * cos(phi1)) * sin(theta)), ((cos(delta) * (1.0 - (0.5 + (-0.5 * cos((phi1 * 2.0)))))) - (cos(phi1) * (sin(delta) * sin(phi1)))));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[delta], $MachinePrecision] * N[(1.0 - N[(0.5 + N[(-0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\left(\cos delta - \left(\cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right) + \cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)}\right)\right) \]
8. Step-by-step derivation

   1. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\left(\cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right) - \color{blue}{\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\left(\cos delta - \cos delta \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)\right), \color{blue}{\left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}\right)\right)\right) \]

9. Simplified 95.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta}{\color{blue}{\left(\left(0 - \left(0.5 + \cos \left(\phi_1 \cdot 2\right) \cdot -0.5\right)\right) + 1\right) \cdot \cos delta - \cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)}} \]

10. Final simplification 95.1%

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

Alternative 6: 91.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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(delta) * sin(theta))), (cos(delta) + ((0.5d0 * cos((phi1 * 2.0d0))) - 0.5d0)))
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(delta) * Math.sin(theta))), (Math.cos(delta) + ((0.5 * Math.cos((phi1 * 2.0))) - 0.5)));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), (math.cos(delta) + ((0.5 * math.cos((phi1 * 2.0))) - 0.5)))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) + ((0.5 * cos((phi1 * 2.0))) - 0.5)));
end

Wolfram

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

Derivation

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 delta around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \color{blue}{\left({\sin \phi_1}^{2}\right)}\right)\right)\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\sin \phi_1, \color{blue}{2}\right)\right)\right)\right) \]

   2. sin-lowering-sin.f64 94.7%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), 2\right)\right)\right)\right) \]

5. Simplified 94.7%

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

   1. +-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}}, \color{blue}{\lambda_1}\right) \]

7. Applied egg-rr 94.7%

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

8. Final simplification 94.7%

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

Alternative 7: 91.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -5.7:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 10^{-17}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -5.7)
   (+
    lambda1
    (atan2
     (* (cos phi1) (* (sin delta) (sin theta)))
     (- (cos delta) (* phi1 phi1))))
   (if (<= delta 1e-17)
     (+
      lambda1
      (atan2 (* (sin theta) (* delta (cos phi1))) (pow (cos phi1) 2.0)))
     (+
      lambda1
      (atan2 (* (sin delta) (* (cos phi1) (sin theta))) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -5.7) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (phi1 * phi1)));
	} else if (delta <= 1e-17) {
		tmp = lambda1 + atan2((sin(theta) * (delta * cos(phi1))), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), 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 <= (-5.7d0)) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (phi1 * phi1)))
    else if (delta <= 1d-17) then
        tmp = lambda1 + atan2((sin(theta) * (delta * cos(phi1))), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), 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 <= -5.7) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), (Math.cos(delta) - (phi1 * phi1)));
	} else if (delta <= 1e-17) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (delta * Math.cos(phi1))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.cos(phi1) * Math.sin(theta))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -5.7:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), (math.cos(delta) - (phi1 * phi1)))
	elif delta <= 1e-17:
		tmp = lambda1 + math.atan2((math.sin(theta) * (delta * math.cos(phi1))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.cos(phi1) * math.sin(theta))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -5.7)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(cos(delta) - Float64(phi1 * phi1))));
	elseif (delta <= 1e-17)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(delta * cos(phi1))), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(cos(phi1) * sin(theta))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -5.7)
		tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (phi1 * phi1)));
	elseif (delta <= 1e-17)
		tmp = lambda1 + atan2((sin(theta) * (delta * cos(phi1))), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -5.7], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 1e-17], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -5.70000000000000018

   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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \color{blue}{\left({\sin \phi_1}^{2}\right)}\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\sin \phi_1, \color{blue}{2}\right)\right)\right)\right) \]

      2. sin-lowering-sin.f64 89.1%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), 2\right)\right)\right)\right) \]

   5. Simplified 89.1%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\cos delta + -1 \cdot {\phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \left(\mathsf{neg}\left({\phi_1}^{2}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \color{blue}{{\phi_1}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\cos delta, \color{blue}{\left({\phi_1}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left({\color{blue}{\phi_1}}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\phi_1 \cdot \color{blue}{\phi_1}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 87.3%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\phi_1}\right)\right)\right)\right) \]

   8. Simplified 87.3%

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

   if -5.70000000000000018 < delta < 1.00000000000000007e-17

   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 delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \color{blue}{\left({\sin \phi_1}^{2}\right)}\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\sin \phi_1, \color{blue}{2}\right)\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), 2\right)\right)\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(1 - {\sin \phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(1 - \sin \phi_1 \cdot \color{blue}{\sin \phi_1}\right)\right)\right) \]

      2. 1-sub-sin N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_1}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left({\cos \phi_1}^{\color{blue}{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{pow.f64}\left(\cos \phi_1, \color{blue}{2}\right)\right)\right) \]

      5. cos-lowering-cos.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right)\right) \]

   8. Simplified 99.9%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\phi_1\right)}, 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\phi_1\right)}, 2\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\cos \phi_1 \cdot delta\right), \sin theta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\phi_1}\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, delta\right), \sin theta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\phi_1}\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), delta\right), \sin theta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right)\right) \]

       6. sin-lowering-sin.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right)\right) \]

   11. Simplified 99.9%

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

   if 1.00000000000000007e-17 < 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. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \color{blue}{\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\right), \color{blue}{\left(\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 89.6%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 89.6%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -5.7:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 10^{-17}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \color{blue}{\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\right), \color{blue}{\left(\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \color{blue}{\cos delta}\right)\right) \]
6. Step-by-step derivation

   1. cos-lowering-cos.f64 90.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

7. Simplified 90.8%

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

Alternative 9: 86.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
6. Step-by-step derivation

   1. cos-lowering-cos.f64 90.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

7. Simplified 90.8%

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

8. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   3. sin-lowering-sin.f64 88.3%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

10. Simplified 88.3%

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

Alternative 10: 81.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{-0.5 \cdot \left(delta \cdot delta\right) + 1}\\ \mathbf{if}\;theta \leq -1 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 2.9:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(-0.16666666666666666 \cdot \left(theta \cdot theta\right) + 1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (* (sin delta) (sin theta))
           (+ (* -0.5 (* delta delta)) 1.0)))))
   (if (<= theta -1e-72)
     t_1
     (if (<= theta 2.9)
       (+
        lambda1
        (atan2
         (*
          theta
          (* (sin delta) (+ (* -0.16666666666666666 (* theta theta)) 1.0)))
         (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * sin(theta)), ((-0.5 * (delta * delta)) + 1.0));
	double tmp;
	if (theta <= -1e-72) {
		tmp = t_1;
	} else if (theta <= 2.9) {
		tmp = lambda1 + atan2((theta * (sin(delta) * ((-0.16666666666666666 * (theta * theta)) + 1.0))), cos(delta));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = lambda1 + atan2((sin(delta) * sin(theta)), (((-0.5d0) * (delta * delta)) + 1.0d0))
    if (theta <= (-1d-72)) then
        tmp = t_1
    else if (theta <= 2.9d0) then
        tmp = lambda1 + atan2((theta * (sin(delta) * (((-0.16666666666666666d0) * (theta * theta)) + 1.0d0))), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), ((-0.5 * (delta * delta)) + 1.0));
	double tmp;
	if (theta <= -1e-72) {
		tmp = t_1;
	} else if (theta <= 2.9) {
		tmp = lambda1 + Math.atan2((theta * (Math.sin(delta) * ((-0.16666666666666666 * (theta * theta)) + 1.0))), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), ((-0.5 * (delta * delta)) + 1.0))
	tmp = 0
	if theta <= -1e-72:
		tmp = t_1
	elif theta <= 2.9:
		tmp = lambda1 + math.atan2((theta * (math.sin(delta) * ((-0.16666666666666666 * (theta * theta)) + 1.0))), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), Float64(Float64(-0.5 * Float64(delta * delta)) + 1.0)))
	tmp = 0.0
	if (theta <= -1e-72)
		tmp = t_1;
	elseif (theta <= 2.9)
		tmp = Float64(lambda1 + atan(Float64(theta * Float64(sin(delta) * Float64(Float64(-0.16666666666666666 * Float64(theta * theta)) + 1.0))), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(delta) * sin(theta)), ((-0.5 * (delta * delta)) + 1.0));
	tmp = 0.0;
	if (theta <= -1e-72)
		tmp = t_1;
	elseif (theta <= 2.9)
		tmp = lambda1 + atan2((theta * (sin(delta) * ((-0.16666666666666666 * (theta * theta)) + 1.0))), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[(-0.5 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1e-72], t$95$1, If[LessEqual[theta, 2.9], N[(lambda1 + N[ArcTan[N[(theta * N[(N[Sin[delta], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -9.9999999999999997e-73 or 2.89999999999999991 < theta

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 86.5%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 86.5%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 84.6%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 84.6%

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

   11. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {delta}^{2}\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {delta}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({delta}^{2}\right)}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(delta \cdot \color{blue}{delta}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(delta, \color{blue}{delta}\right)\right)\right)\right)\right) \]

   13. Simplified 77.3%

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

   if -9.9999999999999997e-73 < theta < 2.89999999999999991

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 96.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 96.8%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 93.6%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 93.6%

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

   11. Taylor expanded in theta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(theta \cdot \left(\sin delta + \frac{-1}{6} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(theta \cdot \left(\sin delta + \frac{-1}{6} \cdot \left(\sin delta \cdot {theta}^{2}\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(theta \cdot \left(\sin delta + \left(\frac{-1}{6} \cdot \sin delta\right) \cdot {theta}^{2}\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \left(\sin delta + \left(\frac{-1}{6} \cdot \sin delta\right) \cdot {theta}^{2}\right)\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \left(\sin delta + \frac{-1}{6} \cdot \left(\sin delta \cdot {theta}^{2}\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \left(\sin delta + \frac{-1}{6} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \left(\sin delta + \left(\frac{-1}{6} \cdot {theta}^{2}\right) \cdot \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

       7. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) \cdot \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right), \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot {theta}^{2}\right), 1\right), \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left({theta}^{2}\right)\right), 1\right), \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(theta \cdot theta\right)\right), 1\right), \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(theta, theta\right)\right), 1\right), \sin delta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

       13. sin-lowering-sin.f64 93.8%

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(theta, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(theta, theta\right)\right), 1\right), \mathsf{sin.f64}\left(delta\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   13. Simplified 93.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left(\left(-0.16666666666666666 \cdot \left(theta \cdot theta\right) + 1\right) \cdot \sin delta\right)}}{\cos delta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1 \cdot 10^{-72}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{-0.5 \cdot \left(delta \cdot delta\right) + 1}\\ \mathbf{elif}\;theta \leq 2.9:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(-0.16666666666666666 \cdot \left(theta \cdot theta\right) + 1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{-0.5 \cdot \left(delta \cdot delta\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{-0.5 \cdot \left(delta \cdot delta\right) + 1}\\ \mathbf{if}\;theta \leq -1.35 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (* (sin delta) (sin theta))
           (+ (* -0.5 (* delta delta)) 1.0)))))
   (if (<= theta -1.35e-74)
     t_1
     (if (<= theta 7.5e+38)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * sin(theta)), ((-0.5 * (delta * delta)) + 1.0));
	double tmp;
	if (theta <= -1.35e-74) {
		tmp = t_1;
	} else if (theta <= 7.5e+38) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = lambda1 + atan2((sin(delta) * sin(theta)), (((-0.5d0) * (delta * delta)) + 1.0d0))
    if (theta <= (-1.35d-74)) then
        tmp = t_1
    else if (theta <= 7.5d+38) then
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), ((-0.5 * (delta * delta)) + 1.0));
	double tmp;
	if (theta <= -1.35e-74) {
		tmp = t_1;
	} else if (theta <= 7.5e+38) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), ((-0.5 * (delta * delta)) + 1.0))
	tmp = 0
	if theta <= -1.35e-74:
		tmp = t_1
	elif theta <= 7.5e+38:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), Float64(Float64(-0.5 * Float64(delta * delta)) + 1.0)))
	tmp = 0.0
	if (theta <= -1.35e-74)
		tmp = t_1;
	elseif (theta <= 7.5e+38)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(delta) * sin(theta)), ((-0.5 * (delta * delta)) + 1.0));
	tmp = 0.0;
	if (theta <= -1.35e-74)
		tmp = t_1;
	elseif (theta <= 7.5e+38)
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[(-0.5 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.35e-74], t$95$1, If[LessEqual[theta, 7.5e+38], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -1.35000000000000009e-74 or 7.4999999999999999e38 < theta

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 85.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 85.8%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 83.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 83.8%

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

   11. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {delta}^{2}\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {delta}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({delta}^{2}\right)}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(delta \cdot \color{blue}{delta}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(delta, \color{blue}{delta}\right)\right)\right)\right)\right) \]

   13. Simplified 77.3%

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

   if -1.35000000000000009e-74 < theta < 7.4999999999999999e38

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 97.0%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 97.0%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 94.0%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 94.0%

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

   11. Taylor expanded in theta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(theta \cdot \sin delta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin delta \cdot theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

       3. sin-lowering-sin.f64 92.6%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   13. Simplified 92.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.35 \cdot 10^{-74}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{-0.5 \cdot \left(delta \cdot delta\right) + 1}\\ \mathbf{elif}\;theta \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{-0.5 \cdot \left(delta \cdot delta\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{if}\;theta \leq -0.056:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin delta) (sin theta)) 1.0))))
   (if (<= theta -0.056)
     t_1
     (if (<= theta 2e-30)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * sin(theta)), 1.0);
	double tmp;
	if (theta <= -0.056) {
		tmp = t_1;
	} else if (theta <= 2e-30) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = lambda1 + atan2((sin(delta) * sin(theta)), 1.0d0)
    if (theta <= (-0.056d0)) then
        tmp = t_1
    else if (theta <= 2d-30) then
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), 1.0);
	double tmp;
	if (theta <= -0.056) {
		tmp = t_1;
	} else if (theta <= 2e-30) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), 1.0)
	tmp = 0
	if theta <= -0.056:
		tmp = t_1
	elif theta <= 2e-30:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), 1.0))
	tmp = 0.0
	if (theta <= -0.056)
		tmp = t_1;
	elseif (theta <= 2e-30)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(delta) * sin(theta)), 1.0);
	tmp = 0.0;
	if (theta <= -0.056)
		tmp = t_1;
	elseif (theta <= 2e-30)
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -0.056], t$95$1, If[LessEqual[theta, 2e-30], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -0.0560000000000000012 or 2e-30 < theta

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 85.7%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 85.7%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 83.5%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 83.5%

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

   11. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{1}\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 74.6%

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

   if -0.0560000000000000012 < theta < 2e-30

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 96.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 96.8%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 94.2%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 94.2%

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

   11. Taylor expanded in theta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(theta \cdot \sin delta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin delta \cdot theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

       3. sin-lowering-sin.f64 94.2%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   13. Simplified 94.2%

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

Alternative 13: 81.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -9.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))
   (if (<= delta -9.5)
     t_1
     (if (<= delta 3.1e+16)
       (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta));
	double tmp;
	if (delta <= -9.5) {
		tmp = t_1;
	} else if (delta <= 3.1e+16) {
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta))
    if (delta <= (-9.5d0)) then
        tmp = t_1
    else if (delta <= 3.1d+16) then
        tmp = lambda1 + atan2((delta * sin(theta)), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	double tmp;
	if (delta <= -9.5) {
		tmp = t_1;
	} else if (delta <= 3.1e+16) {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	tmp = 0
	if delta <= -9.5:
		tmp = t_1
	elif delta <= 3.1e+16:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)))
	tmp = 0.0
	if (delta <= -9.5)
		tmp = t_1;
	elseif (delta <= 3.1e+16)
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta));
	tmp = 0.0;
	if (delta <= -9.5)
		tmp = t_1;
	elseif (delta <= 3.1e+16)
		tmp = lambda1 + atan2((delta * sin(theta)), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -9.5], t$95$1, If[LessEqual[delta, 3.1e+16], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -9.5 or 3.1e16 < 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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 87.2%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 87.2%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 83.4%

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

   11. Taylor expanded in theta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(theta \cdot \sin delta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin delta \cdot theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

       3. sin-lowering-sin.f64 74.1%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   13. Simplified 74.1%

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

   if -9.5 < delta < 3.1e16

   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. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 94.3%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   7. Simplified 94.3%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

      3. sin-lowering-sin.f64 93.3%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   10. Simplified 93.3%

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

   11. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

       2. sin-lowering-sin.f64 92.6%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   13. Simplified 92.6%

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

Alternative 14: 75.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{cos.f64}\left(delta\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\color{blue}{delta}\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. sin-lowering-sin.f64 99.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(theta\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \color{blue}{\cos delta}\right)\right) \]
6. Step-by-step derivation

   1. cos-lowering-cos.f64 90.8%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

7. Simplified 90.8%

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

8. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   3. sin-lowering-sin.f64 88.3%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

10. Simplified 88.3%

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

11. Taylor expanded in delta around 0

    \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(delta, \sin theta\right), \mathsf{cos.f64}\left(\color{blue}{delta}\right)\right)\right) \]

    2. sin-lowering-sin.f64 74.3%

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

13. Simplified 74.3%

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

Alternative 15: 71.0% accurate, 1320.0× 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 99.8%

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

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \color{blue}{\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\right), \color{blue}{\left(\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in lambda1 around inf

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

7. Simplified 69.7%

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

Reproduce

herbie shell --seed 2024138 
(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))))))))))