Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.7%

Time: 15.5s

Alternatives: 14

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = atan2((cos(phi1) * (sin(delta) * sin(theta))), ((cos(delta) - ((0.5d0 - (cos((phi1 + phi1)) * 0.5d0)) * cos(delta))) - (sin(phi1) * ((cos(phi1) * sin(delta)) * cos(theta))))) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate--r+ N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1} \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{0}{1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<=
      (atan2
       (* (cos phi1) (* (sin delta) (sin theta)))
       (-
        (cos delta)
        (*
         (sin
          (asin
           (+
            (* (sin phi1) (cos delta))
            (* (* (cos phi1) (sin delta)) (cos theta)))))
         (sin phi1))))
      2e-47)
   (+ (atan2 0.0 1.0) lambda1)
   (+ (atan2 0.0 (fma (* delta delta) -0.5 1.0)) lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))) * sin(phi1)))) <= 2e-47) {
		tmp = atan2(0.0, 1.0) + lambda1;
	} else {
		tmp = atan2(0.0, fma((delta * delta), -0.5, 1.0)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(cos(delta) - Float64(sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))) * sin(phi1)))) <= 2e-47)
		tmp = Float64(atan(0.0, 1.0) + lambda1);
	else
		tmp = Float64(atan(0.0, fma(Float64(delta * delta), -0.5, 1.0)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[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] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-47], N[(N[ArcTan[0.0 / 1.0], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[0.0 / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 1.9999999999999999e-47

   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 94.2

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

   5. Applied rewrites 94.2%

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

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

      5. lift-cos.f64 N/A

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

      6. sin-cos-mult N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower--.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in delta around 0

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

      1. associate-*r* N/A

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

      2. sin-neg N/A

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

      3. unsub-neg N/A

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

      4. +-inverses N/A

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

      5. mul0-rgt 81.7

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

   10. Applied rewrites 81.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

   11. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 82.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{1} \]

   if 1.9999999999999999e-47 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 80.1

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

   5. Applied rewrites 80.1%

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

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

      5. lift-cos.f64 N/A

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

      6. sin-cos-mult N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower--.f64 75.2

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in delta around 0

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

      1. associate-*r* N/A

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

      2. sin-neg N/A

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

      3. unsub-neg N/A

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

      4. +-inverses N/A

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

      5. mul0-rgt 58.2

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

   10. Applied rewrites 58.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

   11. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{1 + \color{blue}{\frac{-1}{2} \cdot {delta}^{2}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 66.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{-0.5}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1} \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{0}{1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = atan2((cos(phi1) * (sin(delta) * sin(theta))), ((cos(delta) - ((0.5d0 - (cos((phi1 + phi1)) * 0.5d0)) * cos(delta))) - ((cos(phi1) * sin(delta)) * sin(phi1)))) + lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate--r+ N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 96.1

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

9. Applied rewrites 96.1%

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

10. Final simplification 96.1%

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

Alternative 6: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 96.0

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

5. Applied rewrites 96.0%

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

6. Final simplification 96.0%

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

Alternative 7: 92.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in delta around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{{\sin \phi_1}^{2}}} \]
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 95.6

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

5. Applied rewrites 95.6%

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

6. Final simplification 95.6%

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

Alternative 8: 89.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.000215:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos \phi_1 \cdot \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -0.000215)
   (+
    (atan2
     (* (cos phi1) (* (sin delta) (sin theta)))
     (* (cos phi1) (cos phi1)))
    lambda1)
   (+ (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta)) lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.000215) {
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) * cos(phi1))) + lambda1;
	} else {
		tmp = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	}
	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 (phi1 <= (-0.000215d0)) then
        tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) * cos(phi1))) + lambda1
    else
        tmp = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.000215) {
		tmp = Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), (Math.cos(phi1) * Math.cos(phi1))) + lambda1;
	} else {
		tmp = Math.atan2(((Math.cos(phi1) * Math.sin(theta)) * Math.sin(delta)), Math.cos(delta)) + lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -0.000215:
		tmp = math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), (math.cos(phi1) * math.cos(phi1))) + lambda1
	else:
		tmp = math.atan2(((math.cos(phi1) * math.sin(theta)) * math.sin(delta)), math.cos(delta)) + lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -0.000215)
		tmp = Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(cos(phi1) * cos(phi1))) + lambda1);
	else
		tmp = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -0.000215)
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) * cos(phi1))) + lambda1;
	else
		tmp = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -0.000215], N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.14999999999999995e-4

   1. Initial program 99.4%

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

   3. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.14999999999999995e-4 < phi1

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 93.5%

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

Alternative 9: 89.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -0.000215], N[(N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(2.0 * phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] / lambda1), $MachinePrecision] * lambda1 + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.14999999999999995e-4

   1. Initial program 99.4%

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

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

      3. lift-sin.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. sin-asin N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. associate--r+ N/A

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

   4. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-+.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in delta around 0

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

   12. Applied rewrites 80.6%

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

   if -2.14999999999999995e-4 < phi1

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 93.4%

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

Alternative 10: 88.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 91.3

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

5. Applied rewrites 91.3%

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

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 91.3

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

7. Applied rewrites 91.3%

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

8. Final simplification 91.3%

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

Alternative 11: 86.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 91.3

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

5. Applied rewrites 91.3%

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

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 90.1

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

8. Applied rewrites 90.1%

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

9. Final simplification 90.1%

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

Alternative 12: 80.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -3050000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)))
   (if (<= delta -3050000000.0)
     t_1
     (if (<= delta 2e+29)
       (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -3050000000.0) {
		tmp = t_1;
	} else if (delta <= 2e+29) {
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1
    if (delta <= (-3050000000.0d0)) then
        tmp = t_1
    else if (delta <= 2d+29) then
        tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -3050000000.0) {
		tmp = t_1;
	} else if (delta <= 2e+29) {
		tmp = Math.atan2((delta * Math.sin(theta)), Math.cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -3050000000.0:
		tmp = t_1
	elif delta <= 2e+29:
		tmp = math.atan2((delta * math.sin(theta)), math.cos(delta)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -3050000000.0)
		tmp = t_1;
	elseif (delta <= 2e+29)
		tmp = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -3050000000.0)
		tmp = t_1;
	elseif (delta <= 2e+29)
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -3050000000.0], t$95$1, If[LessEqual[delta, 2e+29], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.05e9 or 1.99999999999999983e29 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 87.1

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

   5. Applied rewrites 87.1%

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

   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 84.8

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

   8. Applied rewrites 84.8%

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

   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 75.3%

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

   if -3.05e9 < delta < 1.99999999999999983e29

   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 94.0

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

   5. Applied rewrites 94.0%

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

   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 93.6

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

   8. Applied rewrites 93.6%

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

   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 92.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3050000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} + \lambda_1\\ \mathbf{if}\;theta \leq -840:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 15.5:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 0.0 (fma (* delta delta) -0.5 1.0)) lambda1)))
   (if (<= theta -840.0)
     t_1
     (if (<= theta 15.5)
       (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(0.0, fma((delta * delta), -0.5, 1.0)) + lambda1;
	double tmp;
	if (theta <= -840.0) {
		tmp = t_1;
	} else if (theta <= 15.5) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(0.0, fma(Float64(delta * delta), -0.5, 1.0)) + lambda1)
	tmp = 0.0
	if (theta <= -840.0)
		tmp = t_1;
	elseif (theta <= 15.5)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[0.0 / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -840.0], t$95$1, If[LessEqual[theta, 15.5], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -840 or 15.5 < theta

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.0

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

   5. Applied rewrites 89.0%

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

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

      5. lift-cos.f64 N/A

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

      6. sin-cos-mult N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower--.f64 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in delta around 0

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

      1. associate-*r* N/A

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

      2. sin-neg N/A

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

      3. unsub-neg N/A

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

      4. +-inverses N/A

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

      5. mul0-rgt 70.3

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

   10. Applied rewrites 70.3%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

   11. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{1 + \color{blue}{\frac{-1}{2} \cdot {delta}^{2}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 71.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{-0.5}, 1\right)} \]

   if -840 < theta < 15.5

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 93.8

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

   5. Applied rewrites 93.8%

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

   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 92.8

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

   8. Applied rewrites 92.8%

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

   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 92.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -840:\\ \;\;\;\;\tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} + \lambda_1\\ \mathbf{elif}\;theta \leq 15.5:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{0}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{0}{1} + \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ (atan2 0.0 1.0) lambda1))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return atan2(0.0, 1.0) + lambda1;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = atan2(0.0d0, 1.0d0) + lambda1
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return Math.atan2(0.0, 1.0) + lambda1;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return math.atan2(0.0, 1.0) + lambda1

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(0.0, 1.0) + lambda1)
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = atan2(0.0, 1.0) + lambda1;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[0.0 / 1.0], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 91.3

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

5. Applied rewrites 91.3%

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

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

   5. lift-cos.f64 N/A

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

   6. sin-cos-mult N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower--.f64 87.9

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

7. Applied rewrites 87.9%

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

8. Taylor expanded in delta around 0

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

   1. associate-*r* N/A

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

   2. sin-neg N/A

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

   3. unsub-neg N/A

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

   4. +-inverses N/A

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

   5. mul0-rgt 76.9

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

10. Applied rewrites 76.9%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{0}}{\cos delta} \]

11. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{1} \]
12. Step-by-step derivation

13. Applied rewrites 75.4%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{1} \]

14. Final simplification 75.4%

    \[\leadsto \tan^{-1}_* \frac{0}{1} + \lambda_1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))))))))