Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× 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]

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. Add Preprocessing

Alternative 2: 31.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (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 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 = 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 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 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 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 = 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[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]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in lambda1 around 0

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

5. Applied rewrites 27.7%

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

Alternative 3: 10.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -7 \cdot 10^{-54}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos delta - \sin^{-1} \left(\cos \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1))))
   (if (<= delta -7e-54)
     (atan2
      t_1
      (-
       (cos delta)
       (asin
        (+
         (* (cos phi1) (cos delta))
         (* (* (cos phi1) (sin delta)) (cos theta))))))
     (atan2 t_1 (cos phi1)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double tmp;
	if (delta <= -7e-54) {
		tmp = atan2(t_1, (cos(delta) - asin(((cos(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))));
	} else {
		tmp = atan2(t_1, cos(phi1));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (sin(theta) * sin(delta)) * cos(phi1)
    if (delta <= (-7d-54)) then
        tmp = atan2(t_1, (cos(delta) - asin(((cos(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))
    else
        tmp = atan2(t_1, cos(phi1))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1);
	double tmp;
	if (delta <= -7e-54) {
		tmp = Math.atan2(t_1, (Math.cos(delta) - Math.asin(((Math.cos(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))));
	} else {
		tmp = Math.atan2(t_1, Math.cos(phi1));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = (math.sin(theta) * math.sin(delta)) * math.cos(phi1)
	tmp = 0
	if delta <= -7e-54:
		tmp = math.atan2(t_1, (math.cos(delta) - math.asin(((math.cos(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))
	else:
		tmp = math.atan2(t_1, math.cos(phi1))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	tmp = 0.0
	if (delta <= -7e-54)
		tmp = atan(t_1, Float64(cos(delta) - asin(Float64(Float64(cos(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))));
	else
		tmp = atan(t_1, cos(phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	tmp = 0.0;
	if (delta <= -7e-54)
		tmp = atan2(t_1, (cos(delta) - asin(((cos(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))));
	else
		tmp = atan2(t_1, cos(phi1));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -7e-54], N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[ArcSin[N[(N[(N[Cos[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], N[ArcTan[t$95$1 / N[Cos[phi1], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if delta < -6.99999999999999964e-54

   1. Initial program 99.7%

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

   3. Taylor expanded in lambda1 around 0

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 19.8%

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

   9. Taylor expanded in phi1 around 0

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

   11. Applied rewrites 17.4%

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

   if -6.99999999999999964e-54 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in lambda1 around 0

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

   5. Applied rewrites 22.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 10.6%

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

Alternative 4: 14.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (atan2
  (* (* (sin theta) (sin delta)) (cos phi1))
  (-
   (cos delta)
   (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 atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - 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 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - 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 Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) - 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 math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - 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 atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - 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 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in lambda1 around 0

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

5. Applied rewrites 27.7%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 13.7%

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

Alternative 5: 11.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[phi1], $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 lambda1 around 0

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

5. Applied rewrites 27.7%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 10.4%

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

Alternative 6: 10.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1} \]
6. Add Preprocessing

Alternative 7: 9.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \sin theta \cdot \sin delta \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $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 lambda1 around inf

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

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\sin theta \cdot \sin delta} \]
6. Add Preprocessing

Alternative 8: 5.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \sin delta \end{array} \]

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return sin(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 = sin(delta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[Sin[delta], $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 lambda1 around inf

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

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\sin theta \cdot \sin delta} \]

6. Taylor expanded in lambda1 around inf

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

8. Applied rewrites 5.0%

   \[\leadsto \color{blue}{\sin delta} \]
9. Add Preprocessing

Alternative 9: 4.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \cos \phi_1 \end{array} \]

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return cos(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 = cos(phi1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[Cos[phi1], $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 lambda1 around -inf

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos delta + \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{2} \cdot \left(\cos theta \cdot \sin delta\right) + \frac{-1}{6} \cdot \left(\cos theta \cdot \sin delta\right)\right)\right) - \cos delta\right) - \cos theta \cdot \sin delta\right)} \]

8. Applied rewrites 4.6%

   \[\leadsto \cos \phi_1 \]
9. Add Preprocessing

Alternative 10: 4.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \cos delta \end{array} \]

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return 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 = cos(delta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[Cos[delta], $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 lambda1 around inf

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

5. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \sin delta \cdot \sin theta + \color{blue}{{\phi_1}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sin delta \cdot \sin theta\right) + {\phi_1}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\phi_1}^{2} \cdot \left(\sin delta \cdot \sin theta\right)\right) + \frac{1}{24} \cdot \left(\sin delta \cdot \sin theta\right)\right)\right)} \]

8. Applied rewrites 4.2%

   \[\leadsto \cos delta \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))))))))))