Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.8%

Time: 13.6s

Alternatives: 15

Speedup: 1.2×

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: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. unpow2 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-pow.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. unpow2 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-pow.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 99.8%

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

Alternative 5: 94.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - fma(Float64(cos(phi1) * sin(delta)), sin(phi1), Float64(cos(delta) * (sin(phi1) ^ 2.0))))))
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[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 95.4

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

5. Applied rewrites 95.4%

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

7. Applied rewrites 95.4%

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

Alternative 6: 94.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(fma(sin(phi1), cos(delta), Float64(sin(delta) * cos(phi1))) * sin(phi1)))))
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[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 95.4

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

5. Applied rewrites 95.4%

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

Alternative 7: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ \mathbf{if}\;theta \leq -1.3 \cdot 10^{+64} \lor \neg \left(theta \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, -0.5\right) + 1, \cos delta, \left(\left(\sin \phi_1 \cdot \cos \phi_1\right) \cdot \sin delta\right) \cdot \mathsf{fma}\left(0.5 \cdot theta, theta, -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1))))
   (if (or (<= theta -1.3e+64) (not (<= theta 3.5e-11)))
     (+ lambda1 (atan2 t_1 (- (cos delta) (pow (sin phi1) 2.0))))
     (+
      lambda1
      (atan2
       t_1
       (fma
        (+ (fma (cos (* 2.0 phi1)) 0.5 -0.5) 1.0)
        (cos delta)
        (*
         (* (* (sin phi1) (cos phi1)) (sin delta))
         (fma (* 0.5 theta) theta -1.0))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double tmp;
	if ((theta <= -1.3e+64) || !(theta <= 3.5e-11)) {
		tmp = lambda1 + atan2(t_1, (cos(delta) - pow(sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + atan2(t_1, fma((fma(cos((2.0 * phi1)), 0.5, -0.5) + 1.0), cos(delta), (((sin(phi1) * cos(phi1)) * sin(delta)) * fma((0.5 * theta), theta, -1.0))));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	tmp = 0.0
	if ((theta <= -1.3e+64) || !(theta <= 3.5e-11))
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(delta) - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(Float64(fma(cos(Float64(2.0 * phi1)), 0.5, -0.5) + 1.0), cos(delta), Float64(Float64(Float64(sin(phi1) * cos(phi1)) * sin(delta)) * fma(Float64(0.5 * theta), theta, -1.0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[theta, -1.3e+64], N[Not[LessEqual[theta, 3.5e-11]], $MachinePrecision]], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[(N[Cos[N[(2.0 * phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * theta), $MachinePrecision] * theta + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if theta < -1.29999999999999998e64 or 3.50000000000000019e-11 < 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. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -1.29999999999999998e64 < theta < 3.50000000000000019e-11

   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. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. unpow2 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in theta around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

       3. associate-+r+ N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt1-in N/A

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

       6. lower-fma.f64 N/A

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

   11. Applied rewrites 99.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.3 \cdot 10^{+64} \lor \neg \left(theta \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, -0.5\right) + 1, \cos delta, \left(\left(\sin \phi_1 \cdot \cos \phi_1\right) \cdot \sin delta\right) \cdot \mathsf{fma}\left(0.5 \cdot theta, theta, -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -8.2 \cdot 10^{+26} \lor \neg \left(theta \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta - \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \cos \phi_1\right) \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -8.2e+26) (not (<= theta 3.5e-11)))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (- (cos delta) (pow (sin phi1) 2.0))))
   (+
    lambda1
    (atan2
     (* (* theta (cos phi1)) (sin delta))
     (-
      (cos delta)
      (*
       (fma (sin phi1) (cos delta) (* (sin delta) (cos phi1)))
       (sin phi1)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -8.2e+26) || !(theta <= 3.5e-11)) {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - pow(sin(phi1), 2.0)));
	} else {
		tmp = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), (cos(delta) - (fma(sin(phi1), cos(delta), (sin(delta) * cos(phi1))) * sin(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -8.2e+26) || !(theta <= 3.5e-11))
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(theta * cos(phi1)) * sin(delta)), Float64(cos(delta) - Float64(fma(sin(phi1), cos(delta), Float64(sin(delta) * cos(phi1))) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -8.2e+26], N[Not[LessEqual[theta, 3.5e-11]], $MachinePrecision]], 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[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(theta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -8.19999999999999967e26 or 3.50000000000000019e-11 < 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. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -8.19999999999999967e26 < theta < 3.50000000000000019e-11

   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 theta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 99.7

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

   8. Applied rewrites 99.7%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -8.2 \cdot 10^{+26} \lor \neg \left(theta \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta - \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \cos \phi_1\right) \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) ** 2.0d0)))
end function

Java

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

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.pow(math.sin(phi1), 2.0)))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) ^ 2.0)));
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[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 92.1

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

5. Applied rewrites 92.1%

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

Alternative 10: 88.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 88.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 88.7

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

7. Applied rewrites 88.7%

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

Alternative 11: 86.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

6. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 85.8

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

8. Applied rewrites 85.8%

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

Alternative 12: 79.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot delta\\ \mathbf{if}\;theta \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{elif}\;theta \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot t\_1}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) delta)))
   (if (<= theta -2.25e-11)
     (+ lambda1 (atan2 t_1 (cos delta)))
     (if (<= theta 6.8e-46)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       (+
        lambda1
        (atan2
         (* (fma (* -0.16666666666666666 delta) delta 1.0) t_1)
         (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * delta;
	double tmp;
	if (theta <= -2.25e-11) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else if (theta <= 6.8e-46) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = lambda1 + atan2((fma((-0.16666666666666666 * delta), delta, 1.0) * t_1), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * delta)
	tmp = 0.0
	if (theta <= -2.25e-11)
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	elseif (theta <= 6.8e-46)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * t_1), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]}, If[LessEqual[theta, -2.25e-11], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[theta, 6.8e-46], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if theta < -2.25e-11

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 87.5

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

   8. Applied rewrites 87.5%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 79.3%

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

   if -2.25e-11 < theta < 6.79999999999999992e-46

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 85.5

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 85.5%

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

   if 6.79999999999999992e-46 < 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. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 84.9

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 72.1%

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

Alternative 13: 79.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -2.25 \cdot 10^{-11} \lor \neg \left(theta \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -2.25e-11) (not (<= theta 3.5e-11)))
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -2.25e-11) || !(theta <= 3.5e-11)) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * 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 ((theta <= (-2.25d-11)) .or. (.not. (theta <= 3.5d-11))) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = lambda1 + atan2((sin(delta) * 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 ((theta <= -2.25e-11) || !(theta <= 3.5e-11)) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (theta <= -2.25e-11) or not (theta <= 3.5e-11):
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -2.25e-11) || !(theta <= 3.5e-11))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((theta <= -2.25e-11) || ~((theta <= 3.5e-11)))
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -2.25e-11], N[Not[LessEqual[theta, 3.5e-11]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -2.25e-11 or 3.50000000000000019e-11 < theta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 86.8

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.6%

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

   if -2.25e-11 < theta < 3.50000000000000019e-11

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 84.7

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 84.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2.25 \cdot 10^{-11} \lor \neg \left(theta \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -9.2 \cdot 10^{+64} \lor \neg \left(theta \leq 4.5 \cdot 10^{+108}\right):\\ \;\;\;\;\left(-\lambda_1\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -9.2e+64) (not (<= theta 4.5e+108)))
   (* (- lambda1) -1.0)
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -9.2e+64) || !(theta <= 4.5e+108)) {
		tmp = -lambda1 * -1.0;
	} else {
		tmp = lambda1 + atan2((sin(delta) * 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 ((theta <= (-9.2d+64)) .or. (.not. (theta <= 4.5d+108))) then
        tmp = -lambda1 * (-1.0d0)
    else
        tmp = lambda1 + atan2((sin(delta) * 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 ((theta <= -9.2e+64) || !(theta <= 4.5e+108)) {
		tmp = -lambda1 * -1.0;
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (theta <= -9.2e+64) or not (theta <= 4.5e+108):
		tmp = -lambda1 * -1.0
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -9.2e+64) || !(theta <= 4.5e+108))
		tmp = Float64(Float64(-lambda1) * -1.0);
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((theta <= -9.2e+64) || ~((theta <= 4.5e+108)))
		tmp = -lambda1 * -1.0;
	else
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -9.2e+64], N[Not[LessEqual[theta, 4.5e+108]], $MachinePrecision]], N[((-lambda1) * -1.0), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -9.2e64 or 4.5e108 < 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. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in lambda1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in lambda1 around inf

      \[\leadsto \left(-\lambda_1\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto \left(-\lambda_1\right) \cdot -1 \]

   if -9.2e64 < theta < 4.5e108

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 84.2

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

   8. Applied rewrites 84.2%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 79.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -9.2 \cdot 10^{+64} \lor \neg \left(theta \leq 4.5 \cdot 10^{+108}\right):\\ \;\;\;\;\left(-\lambda_1\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.3% accurate, 167.6× speedup

Math

\[\begin{array}{l} \\ \left(-\lambda_1\right) \cdot -1 \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (* (- lambda1) -1.0))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return -lambda1 * -1.0;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = -lambda1 * (-1.0d0)
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return -lambda1 * -1.0;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return -lambda1 * -1.0

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(Float64(-lambda1) * -1.0)
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = -lambda1 * -1.0;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[((-lambda1) * -1.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.7

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

5. Applied rewrites 88.7%

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

6. Taylor expanded in lambda1 around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in lambda1 around inf

   \[\leadsto \left(-\lambda_1\right) \cdot -1 \]
10. Step-by-step derivation

11. Applied rewrites 67.1%

    \[\leadsto \left(-\lambda_1\right) \cdot -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024309 
(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))))))))))