Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 13.3s

Alternatives: 19

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: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.7%

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

Alternative 2: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ t_2 := t\_1 \cdot \cos \phi_1\\ \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 3.14159265:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta))) (t_2 (* t_1 (cos phi1))))
   (if (<=
        (+
         lambda1
         (atan2
          t_2
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (sin phi1) (cos delta))
               (* (* (cos phi1) (sin delta)) (cos theta)))))))))
        3.14159265)
     (+ lambda1 (atan2 t_1 (cos delta)))
     (+ lambda1 (atan2 t_2 (fma (* delta delta) -0.5 1.0))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double t_2 = t_1 * cos(phi1);
	double tmp;
	if ((lambda1 + atan2(t_2, (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))) <= 3.14159265) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else {
		tmp = lambda1 + atan2(t_2, fma((delta * delta), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	t_2 = Float64(t_1 * cos(phi1))
	tmp = 0.0
	if (Float64(lambda1 + atan(t_2, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))))) <= 3.14159265)
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_2, fma(Float64(delta * delta), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 + N[ArcTan[t$95$2 / 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], 3.14159265], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 3.14159265000000021

   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 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 85.1

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

   8. Applied rewrites 85.1%

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

   if 3.14159265000000021 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in delta around 0

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

   8. Applied rewrites 100.0%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. 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. fp-cancel-sub-sign-inv N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.7%

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

Alternative 5: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

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) * fma(sin(phi1), cos(delta), (cos(theta) * (cos(phi1) * sin(delta)))))));
}

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) * fma(sin(phi1), cos(delta), Float64(cos(theta) * Float64(cos(phi1) * sin(delta))))))))
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. sin-asin 99.7

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 6: 94.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 95.7

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

7. Applied rewrites 95.7%

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

Alternative 7: 94.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. 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. fp-cancel-sub-sign-inv N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 95.7

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

7. Applied rewrites 95.7%

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

Alternative 8: 94.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 95.6

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

5. Applied rewrites 95.6%

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

Alternative 9: 92.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in delta around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{{\sin \phi_1}^{2}}} \]
6. 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 93.5

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

7. Applied rewrites 93.5%

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

9. Applied rewrites 93.5%

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

Alternative 10: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.00085 \lor \neg \left(delta \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -0.00085) (not (<= delta 1.8e-7)))
   (+ lambda1 (atan2 (* (* (sin theta) (cos phi1)) (sin delta)) (cos delta)))
   (+
    lambda1
    (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (pow (cos phi1) 2.0)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -0.00085) || !(delta <= 1.8e-7)) {
		tmp = lambda1 + atan2(((sin(theta) * cos(phi1)) * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), pow(cos(phi1), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-0.00085d0)) .or. (.not. (delta <= 1.8d-7))) then
        tmp = lambda1 + atan2(((sin(theta) * cos(phi1)) * sin(delta)), cos(delta))
    else
        tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(phi1) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -0.00085) || !(delta <= 1.8e-7)) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.cos(phi1)) * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), Math.pow(Math.cos(phi1), 2.0));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -0.00085) or not (delta <= 1.8e-7):
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.cos(phi1)) * math.sin(delta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), math.pow(math.cos(phi1), 2.0))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -0.00085) || !(delta <= 1.8e-7))
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * cos(phi1)) * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), (cos(phi1) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -0.00085) || ~((delta <= 1.8e-7)))
		tmp = lambda1 + atan2(((sin(theta) * cos(phi1)) * sin(delta)), cos(delta));
	else
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(phi1) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -0.00085], N[Not[LessEqual[delta, 1.8e-7]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -8.49999999999999953e-4 or 1.79999999999999997e-7 < delta

   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 87.5

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

   5. Applied rewrites 87.5%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta}} \]
   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 87.5

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

   7. Applied rewrites 87.5%

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

   if -8.49999999999999953e-4 < delta < 1.79999999999999997e-7

   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

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{1 - {\sin \phi_1}^{2}}} \]
   6. 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-rev 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. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.00085 \lor \neg \left(delta \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.8

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

5. Applied rewrites 89.8%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta}} \]
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 89.8

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

7. Applied rewrites 89.8%

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

Alternative 12: 86.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.8

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

5. Applied rewrites 89.8%

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

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 87.6

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

8. Applied rewrites 87.6%

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

Alternative 13: 81.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -5.4 \lor \neg \left(theta \leq 2.9 \cdot 10^{+31}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot theta\right) \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -5.4) (not (<= theta 2.9e+31)))
   (+
    lambda1
    (atan2 (* (sin theta) (sin delta)) (fma (* delta delta) -0.5 1.0)))
   (+
    lambda1
    (atan2
     (* (* (fma (* theta theta) -0.16666666666666666 1.0) theta) (sin delta))
     (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -5.4) || !(theta <= 2.9e+31)) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), fma((delta * delta), -0.5, 1.0));
	} else {
		tmp = lambda1 + atan2(((fma((theta * theta), -0.16666666666666666, 1.0) * theta) * sin(delta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -5.4) || !(theta <= 2.9e+31))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), fma(Float64(delta * delta), -0.5, 1.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(fma(Float64(theta * theta), -0.16666666666666666, 1.0) * theta) * sin(delta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -5.4], N[Not[LessEqual[theta, 2.9e+31]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[(N[(theta * theta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * theta), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -5.4000000000000004 or 2.9e31 < 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 84.8

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

   5. Applied rewrites 84.8%

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

   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 83.5

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 76.1%

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

   if -5.4000000000000004 < theta < 2.9e31

   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 94.1

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

   5. Applied rewrites 94.1%

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

   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 91.1

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

   8. Applied rewrites 91.1%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 91.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -5.4 \lor \neg \left(theta \leq 2.9 \cdot 10^{+31}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot theta\right) \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -3.7 \cdot 10^{+20} \lor \neg \left(theta \leq 0.065\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -3.7e+20) (not (<= theta 0.065)))
   (+
    lambda1
    (atan2 (* (sin theta) (sin delta)) (fma (* delta delta) -0.5 1.0)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -3.7e+20) || !(theta <= 0.065)) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), fma((delta * delta), -0.5, 1.0));
	} else {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -3.7e+20) || !(theta <= 0.065))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), fma(Float64(delta * delta), -0.5, 1.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -3.7e+20], N[Not[LessEqual[theta, 0.065]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -3.7e20 or 0.065000000000000002 < 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 84.3

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

   5. Applied rewrites 84.3%

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

   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 82.6

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.4%

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

   if -3.7e20 < theta < 0.065000000000000002

   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 91.8

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

   8. Applied rewrites 91.8%

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

   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 91.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -3.7 \cdot 10^{+20} \lor \neg \left(theta \leq 0.065\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -7.6 \lor \neg \left(theta \leq 0.32\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -7.6) (not (<= theta 0.32)))
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -7.6) || !(theta <= 0.32)) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((theta <= (-7.6d0)) .or. (.not. (theta <= 0.32d0))) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -7.6) || !(theta <= 0.32)) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (theta <= -7.6) or not (theta <= 0.32):
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -7.6) || !(theta <= 0.32))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((theta <= -7.6) || ~((theta <= 0.32)))
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -7.6], N[Not[LessEqual[theta, 0.32]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -7.5999999999999996 or 0.320000000000000007 < 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 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 82.7

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

   8. Applied rewrites 82.7%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.1%

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

   if -7.5999999999999996 < theta < 0.320000000000000007

   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 95.1

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

   5. Applied rewrites 95.1%

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

   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 92.2

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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 92.2%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -7.6 \lor \neg \left(theta \leq 0.32\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.8

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

5. Applied rewrites 89.8%

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

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 87.6

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

8. Applied rewrites 87.6%

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

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 74.7%

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

Alternative 17: 69.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(delta \cdot delta\right) - 0.5, delta \cdot delta, 1\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin delta) theta)
   (fma
    (-
     (*
      (fma (* delta delta) -0.001388888888888889 0.041666666666666664)
      (* delta delta))
     0.5)
    (* delta delta)
    1.0))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(delta) * theta), fma(((fma((delta * delta), -0.001388888888888889, 0.041666666666666664) * (delta * delta)) - 0.5), (delta * delta), 1.0));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(delta) * theta), fma(Float64(Float64(fma(Float64(delta * delta), -0.001388888888888889, 0.041666666666666664) * Float64(delta * delta)) - 0.5), Float64(delta * delta), 1.0)))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[(N[(N[(N[(N[(delta * delta), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(delta * delta), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.8

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

5. Applied rewrites 89.8%

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

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 87.6

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

8. Applied rewrites 87.6%

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

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 74.7%

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

12. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{1 + \color{blue}{{delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right)}} \]
13. Step-by-step derivation

14. Applied rewrites 70.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(delta \cdot delta\right) - 0.5, \color{blue}{delta \cdot delta}, 1\right)} \]
15. Add Preprocessing

Alternative 18: 69.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.8

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

5. Applied rewrites 89.8%

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

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 87.6

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

8. Applied rewrites 87.6%

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

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 74.7%

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

12. Taylor expanded in delta around 0

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

14. Applied rewrites 70.0%

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

Alternative 19: 67.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.8

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

5. Applied rewrites 89.8%

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

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 87.6

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

8. Applied rewrites 87.6%

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

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 74.7%

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

12. Taylor expanded in delta around 0

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

14. Applied rewrites 65.2%

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

Reproduce

herbie shell --seed 2024339 
(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))))))))))