Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 16

Speedup: 0.9×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 99.7

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

3. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-asin N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sin-PI/2 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in phi1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 99.8

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

10. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (fma
      (cos phi1)
      (* (cos theta) (sin 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)), (cos(delta) - (sin(phi1) * fma(cos(phi1), (cos(theta) * sin(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)), Float64(cos(delta) - Float64(sin(phi1) * fma(cos(phi1), Float64(cos(theta) * sin(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[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

   4. lift-*.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-asin N/A

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

   13. +-commutative N/A

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

3. Applied rewrites 99.8%

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

Alternative 3: 94.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (* (sin phi1) (fma (sin delta) (cos phi1) (* (cos 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) - (sin(phi1) * fma(sin(delta), cos(phi1), (cos(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) - Float64(sin(phi1) * fma(sin(delta), cos(phi1), Float64(cos(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[Sin[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. sin-sum-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-+.f64 92.0

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

4. Applied rewrites 92.0%

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

   1. lift-+.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. +-commutative N/A

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

   4. sin-sum N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sin.f64 94.6

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

6. Applied rewrites 94.6%

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

Alternative 4: 92.0% accurate, 1.4× speedup

Math

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

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)))))))

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)))));
}

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), 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[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. sin-sum-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-+.f64 92.0

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

4. Applied rewrites 92.0%

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

   1. lift-+.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. sin-sum N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 94.6

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

6. Applied rewrites 94.6%

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

7. Taylor expanded in phi1 around 0

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

   1. lift-sin.f64 91.6

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

9. Applied rewrites 91.6%

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

Alternative 5: 92.0% accurate, 1.8× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    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((phi1 + 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(phi1)), (Math.cos(delta) - (Math.sin(phi1) * Math.sin((phi1 + delta)))));
}

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((phi1 + 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) * sin(Float64(phi1 + delta))))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. sin-sum-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-+.f64 92.0

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

4. Applied rewrites 92.0%

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

Alternative 6: 91.6% accurate, 2.0× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    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((2.0d0 * 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((2.0 * 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((2.0 * 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(2.0 * 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((2.0 * 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[(2.0 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in delta around 0

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

   1. unpow2 N/A

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

   2. sqr-sin-a N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 92.0

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

4. Applied rewrites 92.0%

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

Alternative 7: 88.5% accurate, 2.6× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    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))
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));
}

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(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[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in phi1 around 0

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

   1. lift-cos.f64 88.5

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

4. Applied rewrites 88.5%

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

Alternative 8: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -0.066:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+ lambda1 (atan2 (* (* theta (cos phi1)) (sin delta)) (cos delta)))))
   (if (<= delta -0.066)
     t_1
     (if (<= delta 2e-40)
       (+
        lambda1
        (atan2
         (*
          (*
           (sin theta)
           (* delta (fma -0.16666666666666666 (* delta delta) 1.0)))
          (cos phi1))
         (fma (cos (+ phi1 phi1)) 0.5 0.5)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -0.066) {
		tmp = t_1;
	} else if (delta <= 2e-40) {
		tmp = lambda1 + atan2(((sin(theta) * (delta * fma(-0.16666666666666666, (delta * delta), 1.0))) * cos(phi1)), fma(cos((phi1 + phi1)), 0.5, 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(Float64(theta * cos(phi1)) * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -0.066)
		tmp = t_1;
	elseif (delta <= 2e-40)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * Float64(delta * fma(-0.16666666666666666, Float64(delta * delta), 1.0))) * cos(phi1)), fma(cos(Float64(phi1 + phi1)), 0.5, 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[(theta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.066], t$95$1, If[LessEqual[delta, 2e-40], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -0.066000000000000003 or 1.9999999999999999e-40 < 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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if -0.066000000000000003 < delta < 1.9999999999999999e-40

   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. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot \color{blue}{delta}, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right), -0.5, 0.5\right) \cdot delta\right) - \left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1, delta, 1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \color{blue}{\left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right), -0.5, 0.5\right) \cdot delta\right) - \left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1, delta, 1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right)} \]

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\cos \left(2 \cdot \phi_1\right) \cdot \frac{1}{2} + \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)} \]

      4. lift-cos.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)} \]

      5. count-2-rev N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)} \]

      6. lower-+.f64 99.5

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), 0.5, 0.5\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), \color{blue}{0.5}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{if}\;theta \leq -4.7 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 1.25 \cdot 10^{-16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (* (* (sin theta) (sin delta)) (cos phi1))
           (fma -0.5 (* delta delta) 1.0)))))
   (if (<= theta -4.7e+22)
     t_1
     (if (<= theta 1.25e-16)
       (+ lambda1 (atan2 (* (* theta (cos phi1)) (sin delta)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), fma(-0.5, (delta * delta), 1.0));
	double tmp;
	if (theta <= -4.7e+22) {
		tmp = t_1;
	} else if (theta <= 1.25e-16) {
		tmp = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), fma(-0.5, Float64(delta * delta), 1.0)))
	tmp = 0.0
	if (theta <= -4.7e+22)
		tmp = t_1;
	elseif (theta <= 1.25e-16)
		tmp = Float64(lambda1 + atan(Float64(Float64(theta * cos(phi1)) * sin(delta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -4.7e+22], t$95$1, If[LessEqual[theta, 1.25e-16], N[(lambda1 + N[ArcTan[N[(N[(theta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -4.70000000000000021e22 or 1.2500000000000001e-16 < theta

   1. Initial program 99.7%

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

   2. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.1

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

   7. Applied rewrites 74.1%

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

   if -4.70000000000000021e22 < theta < 1.2500000000000001e-16

   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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 10: 81.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -59000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot delta\right) \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+ lambda1 (atan2 (* (* theta (cos phi1)) (sin delta)) (cos delta)))))
   (if (<= delta -59000000.0)
     t_1
     (if (<= delta 2e-40)
       (+ lambda1 (atan2 (* (* (cos phi1) delta) (sin theta)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -59000000.0) {
		tmp = t_1;
	} else if (delta <= 2e-40) {
		tmp = lambda1 + atan2(((cos(phi1) * delta) * sin(theta)), cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), cos(delta))
    if (delta <= (-59000000.0d0)) then
        tmp = t_1
    else if (delta <= 2d-40) then
        tmp = lambda1 + atan2(((cos(phi1) * delta) * sin(theta)), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2(((theta * Math.cos(phi1)) * Math.sin(delta)), Math.cos(delta));
	double tmp;
	if (delta <= -59000000.0) {
		tmp = t_1;
	} else if (delta <= 2e-40) {
		tmp = lambda1 + Math.atan2(((Math.cos(phi1) * delta) * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2(((theta * math.cos(phi1)) * math.sin(delta)), math.cos(delta))
	tmp = 0
	if delta <= -59000000.0:
		tmp = t_1
	elif delta <= 2e-40:
		tmp = lambda1 + math.atan2(((math.cos(phi1) * delta) * math.sin(theta)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(Float64(theta * cos(phi1)) * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -59000000.0)
		tmp = t_1;
	elseif (delta <= 2e-40)
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * delta) * sin(theta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), cos(delta));
	tmp = 0.0;
	if (delta <= -59000000.0)
		tmp = t_1;
	elseif (delta <= 2e-40)
		tmp = lambda1 + atan2(((cos(phi1) * delta) * sin(theta)), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[(theta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -59000000.0], t$95$1, If[LessEqual[delta, 2e-40], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * delta), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -5.9e7 or 1.9999999999999999e-40 < 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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if -5.9e7 < delta < 1.9999999999999999e-40

   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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 77.6

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

   7. Applied rewrites 77.6%

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

   8. Taylor expanded in delta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sin.f64 92.7

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

   10. Applied rewrites 92.7%

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

Alternative 11: 81.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \cos \phi_1\right) \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -1.5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+ lambda1 (atan2 (* (* theta (cos phi1)) (sin delta)) (cos delta)))))
   (if (<= delta -1.5e+27)
     t_1
     (if (<= delta 2e-40)
       (+
        lambda1
        (atan2
         (*
          (*
           (sin theta)
           (* delta (fma -0.16666666666666666 (* delta delta) 1.0)))
          (cos phi1))
         (fma (* delta delta) -0.5 1.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2(((theta * cos(phi1)) * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -1.5e+27) {
		tmp = t_1;
	} else if (delta <= 2e-40) {
		tmp = lambda1 + atan2(((sin(theta) * (delta * fma(-0.16666666666666666, (delta * delta), 1.0))) * cos(phi1)), fma((delta * delta), -0.5, 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(Float64(theta * cos(phi1)) * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -1.5e+27)
		tmp = t_1;
	elseif (delta <= 2e-40)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * Float64(delta * fma(-0.16666666666666666, Float64(delta * delta), 1.0))) * cos(phi1)), fma(Float64(delta * delta), -0.5, 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[(theta * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.5e+27], t$95$1, If[LessEqual[delta, 2e-40], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.49999999999999988e27 or 1.9999999999999999e-40 < 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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if -1.49999999999999988e27 < delta < 1.9999999999999999e-40

   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. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 97.6

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot \color{blue}{delta}, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right), -0.5, 0.5\right) \cdot delta\right) - \left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1, delta, 1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right)} \]

   7. Applied rewrites 97.6%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \color{blue}{\left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right), -0.5, 0.5\right) \cdot delta\right) - \left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1, delta, 1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right)} \]

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\frac{-1}{2} \cdot {delta}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{{delta}^{2} \cdot \frac{-1}{2} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left({delta}^{2}, \frac{-1}{2}, 1\right)} \]

      4. pow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, \frac{-1}{2}, 1\right)} \]

      5. lift-*.f64 91.5

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} \]

   10. Applied rewrites 91.5%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{-0.5}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -4.5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))
   (if (<= delta -4.5e+27)
     t_1
     (if (<= delta 2e-40)
       (+
        lambda1
        (atan2
         (*
          (*
           (sin theta)
           (* delta (fma -0.16666666666666666 (* delta delta) 1.0)))
          (cos phi1))
         (fma (* delta delta) -0.5 1.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -4.5e+27) {
		tmp = t_1;
	} else if (delta <= 2e-40) {
		tmp = lambda1 + atan2(((sin(theta) * (delta * fma(-0.16666666666666666, (delta * delta), 1.0))) * cos(phi1)), fma((delta * delta), -0.5, 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -4.5e+27)
		tmp = t_1;
	elseif (delta <= 2e-40)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * Float64(delta * fma(-0.16666666666666666, Float64(delta * delta), 1.0))) * cos(phi1)), fma(Float64(delta * delta), -0.5, 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -4.5e+27], t$95$1, If[LessEqual[delta, 2e-40], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.4999999999999999e27 or 1.9999999999999999e-40 < 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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 69.2%

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

   if -4.4999999999999999e27 < delta < 1.9999999999999999e-40

   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. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 97.6

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot \color{blue}{delta}, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right), -0.5, 0.5\right) \cdot delta\right) - \left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1, delta, 1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right)} \]

   7. Applied rewrites 97.6%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \color{blue}{\left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right), -0.5, 0.5\right) \cdot delta\right) - \left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1, delta, 1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right)} \]

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\frac{-1}{2} \cdot {delta}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{{delta}^{2} \cdot \frac{-1}{2} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left({delta}^{2}, \frac{-1}{2}, 1\right)} \]

      4. pow2 N/A

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(\frac{-1}{6}, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, \frac{-1}{2}, 1\right)} \]

      5. lift-*.f64 91.5

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)} \]

   10. Applied rewrites 91.5%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \left(delta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)\right) \cdot \cos \phi_1}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{-0.5}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -5.2 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 10^{-202}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))
   (if (<= delta -5.2e-67) t_1 (if (<= delta 1e-202) lambda1 t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -5.2e-67) {
		tmp = t_1;
	} else if (delta <= 1e-202) {
		tmp = lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta))
    if (delta <= (-5.2d-67)) then
        tmp = t_1
    else if (delta <= 1d-202) then
        tmp = lambda1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	double tmp;
	if (delta <= -5.2e-67) {
		tmp = t_1;
	} else if (delta <= 1e-202) {
		tmp = lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	tmp = 0
	if delta <= -5.2e-67:
		tmp = t_1
	elif delta <= 1e-202:
		tmp = lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -5.2e-67)
		tmp = t_1;
	elseif (delta <= 1e-202)
		tmp = lambda1;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta));
	tmp = 0.0;
	if (delta <= -5.2e-67)
		tmp = t_1;
	elseif (delta <= 1e-202)
		tmp = lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -5.2e-67], t$95$1, If[LessEqual[delta, 1e-202], lambda1, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -5.1999999999999998e-67 or 1e-202 < 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. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 70.3%

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

   if -5.1999999999999998e-67 < delta < 1e-202

   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. Taylor expanded in lambda1 around inf

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

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\lambda_1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(delta \cdot delta\right) - 0.5, delta \cdot delta, 1\right)}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (atan2
          (* (* (sin theta) (sin delta)) (cos phi1))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (sin phi1) (cos delta))
               (* (* (cos phi1) (sin delta)) (cos theta)))))))))
        (t_2
         (+
          lambda1
          (atan2
           (* theta (sin delta))
           (fma
            (-
             (*
              (fma (* delta delta) -0.001388888888888889 0.041666666666666664)
              (* delta delta))
             0.5)
            (* delta delta)
            1.0)))))
   (if (<= t_1 -5e-66) t_2 (if (<= t_1 2e-34) lambda1 t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	double t_2 = lambda1 + atan2((theta * sin(delta)), fma(((fma((delta * delta), -0.001388888888888889, 0.041666666666666664) * (delta * delta)) - 0.5), (delta * delta), 1.0));
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_2;
	} else if (t_1 <= 2e-34) {
		tmp = lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = 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))))))))
	t_2 = Float64(lambda1 + atan(Float64(theta * sin(delta)), fma(Float64(Float64(fma(Float64(delta * delta), -0.001388888888888889, 0.041666666666666664) * Float64(delta * delta)) - 0.5), Float64(delta * delta), 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_2;
	elseif (t_1 <= 2e-34)
		tmp = lambda1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $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]}, If[LessEqual[t$95$1, -5e-66], t$95$2, If[LessEqual[t$95$1, 2e-34], lambda1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -4.99999999999999962e-66 or 1.99999999999999986e-34 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 99.7%

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

   2. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 69.6

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 64.3%

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

   11. Taylor expanded in delta around 0

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

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{{delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right) + 1} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right) \cdot {delta}^{2} + 1} \]

       3. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, {delta}^{\color{blue}{2}}, 1\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       5. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       7. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {delta}^{2} + \frac{1}{24}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\left({delta}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left({delta}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       10. pow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, \frac{-1}{720}, \frac{1}{24}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, \frac{-1}{720}, \frac{1}{24}\right) \cdot {delta}^{2} - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       12. pow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(delta \cdot delta\right) - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       13. lift-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(delta \cdot delta\right) - \frac{1}{2}, {delta}^{2}, 1\right)} \]

       14. pow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(\mathsf{fma}\left(delta \cdot delta, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(delta \cdot delta\right) - \frac{1}{2}, delta \cdot delta, 1\right)} \]

       15. lift-*.f64 63.6

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

   13. Applied rewrites 63.6%

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

   if -4.99999999999999962e-66 < (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)))))))) < 1.99999999999999986e-34

   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. Taylor expanded in lambda1 around inf

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

   4. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\lambda_1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\mathsf{fma}\left(delta \cdot delta, -0.5, 1\right)}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (atan2
          (* (* (sin theta) (sin delta)) (cos phi1))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (sin phi1) (cos delta))
               (* (* (cos phi1) (sin delta)) (cos theta)))))))))
        (t_2
         (+
          lambda1
          (atan2 (* theta (sin delta)) (fma (* delta delta) -0.5 1.0)))))
   (if (<= t_1 -5e-66) t_2 (if (<= t_1 2e-34) lambda1 t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	double t_2 = lambda1 + atan2((theta * sin(delta)), fma((delta * delta), -0.5, 1.0));
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_2;
	} else if (t_1 <= 2e-34) {
		tmp = lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = 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))))))))
	t_2 = Float64(lambda1 + atan(Float64(theta * sin(delta)), fma(Float64(delta * delta), -0.5, 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_2;
	elseif (t_1 <= 2e-34)
		tmp = lambda1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-66], t$95$2, If[LessEqual[t$95$1, 2e-34], lambda1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -4.99999999999999962e-66 or 1.99999999999999986e-34 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 99.7%

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

   2. Taylor expanded in theta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 69.6

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 64.3%

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

   11. Taylor expanded in delta around 0

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

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\frac{-1}{2} \cdot {delta}^{2} + 1} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{{delta}^{2} \cdot \frac{-1}{2} + 1} \]

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 63.6

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

   13. Applied rewrites 63.6%

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

   if -4.99999999999999962e-66 < (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)))))))) < 1.99999999999999986e-34

   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. Taylor expanded in lambda1 around inf

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

   4. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\lambda_1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.6% accurate, 396.9× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := lambda1

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in lambda1 around inf

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

4. Applied rewrites 69.6%

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

Reproduce

herbie shell --seed 2025107 
(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))))))))))