Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

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

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

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / (-N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision] + (-N[(N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $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-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.8%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--r+ N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

5. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 - \frac{\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1}{\cos delta - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta}\right) \cdot \left(\cos delta - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta\right)}} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cos-2 N/A

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

   4. sqr-sin-a-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. sub-flip N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 99.7%

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

7. Applied rewrites 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cos-2 N/A

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

   4. sqr-sin-a-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. sub-flip N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 99.7%

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.8%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--r+ N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

5. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 - \frac{\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1}{\cos delta - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta}\right) \cdot \left(\cos delta - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta\right)}} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cos-2 N/A

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

   4. sqr-sin-a-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. sub-flip N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 99.7%

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

7. Applied rewrites 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cos-2 N/A

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

   4. sqr-sin-a-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. sub-flip N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 99.7%

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.8%

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

Alternative 3: 94.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Sin[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. 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. lower-fma.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 94.7%

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

4. Applied rewrites 94.7%

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

Alternative 4: 92.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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. lower-pow.f64 N/A

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

   2. lower-sin.f64 92.6%

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

4. Applied rewrites 92.6%

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

Alternative 5: 91.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -350000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 2.9 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1)))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -350000000000.0)
     t_2
     (if (<= delta 2.9e-40)
       (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0)))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -350000000000.0) {
		tmp = t_2;
	} else if (delta <= 2.9e-40) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (sin(theta) * sin(delta)) * cos(phi1)
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-350000000000.0d0)) then
        tmp = t_2
    else if (delta <= 2.9d-40) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1);
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -350000000000.0) {
		tmp = t_2;
	} else if (delta <= 2.9e-40) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = (math.sin(theta) * math.sin(delta)) * math.cos(phi1)
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -350000000000.0:
		tmp = t_2
	elif delta <= 2.9e-40:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -350000000000.0)
		tmp = t_2;
	elseif (delta <= 2.9e-40)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -350000000000.0)
		tmp = t_2;
	elseif (delta <= 2.9e-40)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -350000000000.0], t$95$2, If[LessEqual[delta, 2.9e-40], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.5e11 or 2.8999999999999999e-40 < delta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   if -3.5e11 < delta < 2.8999999999999999e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 80.9%

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

   6. Applied rewrites 80.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. 1-sub-sin-rev N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 81.0%

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

   8. Applied rewrites 81.0%

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

Alternative 6: 91.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta}\\ \mathbf{if}\;delta \leq -350000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2.9 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), 0.5, 0.5\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (cos delta)))))
   (if (<= delta -350000000000.0)
     t_1
     (if (<= delta 2.9e-40)
       (+
        (atan2
         (* (cos phi1) (* (sin delta) (sin theta)))
         (fma (cos (+ phi1 phi1)) 0.5 0.5))
        lambda1)
       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)), cos(delta));
	double tmp;
	if (delta <= -350000000000.0) {
		tmp = t_1;
	} else if (delta <= 2.9e-40) {
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), fma(cos((phi1 + phi1)), 0.5, 0.5)) + lambda1;
	} 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)), cos(delta)))
	tmp = 0.0
	if (delta <= -350000000000.0)
		tmp = t_1;
	elseif (delta <= 2.9e-40)
		tmp = Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(cos(Float64(phi1 + phi1)), 0.5, 0.5)) + lambda1);
	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[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -350000000000.0], t$95$1, If[LessEqual[delta, 2.9e-40], N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.5e11 or 2.8999999999999999e-40 < delta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   if -3.5e11 < delta < 2.8999999999999999e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 80.9%

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

   6. Applied rewrites 80.9%

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

   8. Applied rewrites 80.9%

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

Alternative 7: 91.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta}\\ \mathbf{if}\;delta \leq -2150000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 2.9 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot delta\right) \cdot \cos \phi_1}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (cos delta)))))
   (if (<= delta -2150000000000.0)
     t_1
     (if (<= delta 2.9e-40)
       (+
        lambda1
        (atan2
         (* (* (sin theta) delta) (cos phi1))
         (- 1.0 (pow (sin phi1) 2.0))))
       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)), cos(delta));
	double tmp;
	if (delta <= -2150000000000.0) {
		tmp = t_1;
	} else if (delta <= 2.9e-40) {
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), (1.0 - pow(sin(phi1), 2.0)));
	} 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(((sin(theta) * sin(delta)) * cos(phi1)), cos(delta))
    if (delta <= (-2150000000000.0d0)) then
        tmp = t_1
    else if (delta <= 2.9d-40) then
        tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), (1.0d0 - (sin(phi1) ** 2.0d0)))
    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(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), Math.cos(delta));
	double tmp;
	if (delta <= -2150000000000.0) {
		tmp = t_1;
	} else if (delta <= 2.9e-40) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * delta) * Math.cos(phi1)), (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), math.cos(delta))
	tmp = 0
	if delta <= -2150000000000.0:
		tmp = t_1
	elif delta <= 2.9e-40:
		tmp = lambda1 + math.atan2(((math.sin(theta) * delta) * math.cos(phi1)), (1.0 - math.pow(math.sin(phi1), 2.0)))
	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)), cos(delta)))
	tmp = 0.0
	if (delta <= -2150000000000.0)
		tmp = t_1;
	elseif (delta <= 2.9e-40)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * delta) * cos(phi1)), Float64(1.0 - (sin(phi1) ^ 2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), cos(delta));
	tmp = 0.0;
	if (delta <= -2150000000000.0)
		tmp = t_1;
	elseif (delta <= 2.9e-40)
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), (1.0 - (sin(phi1) ^ 2.0)));
	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[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2150000000000.0], t$95$1, If[LessEqual[delta, 2.9e-40], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.15e12 or 2.8999999999999999e-40 < delta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   if -2.15e12 < delta < 2.8999999999999999e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 80.9%

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in delta around 0

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

   9. Applied rewrites 77.6%

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

Alternative 8: 89.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{if}\;delta \leq -2.5 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + t\_1\\ \mathbf{elif}\;delta \leq 3.15 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot delta\right) \cdot \cos \phi_1}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_1}{\lambda_1}\right) \cdot \lambda_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (atan2 (* (sin delta) (sin theta)) (cos delta))))
   (if (<= delta -2.5e+16)
     (+ lambda1 t_1)
     (if (<= delta 3.15e-40)
       (+
        lambda1
        (atan2
         (* (* (sin theta) delta) (cos phi1))
         (- 1.0 (pow (sin phi1) 2.0))))
       (* (+ 1.0 (/ t_1 lambda1)) lambda1)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((sin(delta) * sin(theta)), cos(delta));
	double tmp;
	if (delta <= -2.5e+16) {
		tmp = lambda1 + t_1;
	} else if (delta <= 3.15e-40) {
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), (1.0 - pow(sin(phi1), 2.0)));
	} else {
		tmp = (1.0 + (t_1 / lambda1)) * lambda1;
	}
	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 = atan2((sin(delta) * sin(theta)), cos(delta))
    if (delta <= (-2.5d+16)) then
        tmp = lambda1 + t_1
    else if (delta <= 3.15d-40) then
        tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), (1.0d0 - (sin(phi1) ** 2.0d0)))
    else
        tmp = (1.0d0 + (t_1 / lambda1)) * lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2((Math.sin(delta) * Math.sin(theta)), Math.cos(delta));
	double tmp;
	if (delta <= -2.5e+16) {
		tmp = lambda1 + t_1;
	} else if (delta <= 3.15e-40) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * delta) * Math.cos(phi1)), (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	} else {
		tmp = (1.0 + (t_1 / lambda1)) * lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((math.sin(delta) * math.sin(theta)), math.cos(delta))
	tmp = 0
	if delta <= -2.5e+16:
		tmp = lambda1 + t_1
	elif delta <= 3.15e-40:
		tmp = lambda1 + math.atan2(((math.sin(theta) * delta) * math.cos(phi1)), (1.0 - math.pow(math.sin(phi1), 2.0)))
	else:
		tmp = (1.0 + (t_1 / lambda1)) * lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(sin(delta) * sin(theta)), cos(delta))
	tmp = 0.0
	if (delta <= -2.5e+16)
		tmp = Float64(lambda1 + t_1);
	elseif (delta <= 3.15e-40)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * delta) * cos(phi1)), Float64(1.0 - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(t_1 / lambda1)) * lambda1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((sin(delta) * sin(theta)), cos(delta));
	tmp = 0.0;
	if (delta <= -2.5e+16)
		tmp = lambda1 + t_1;
	elseif (delta <= 3.15e-40)
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), (1.0 - (sin(phi1) ^ 2.0)));
	else
		tmp = (1.0 + (t_1 / lambda1)) * lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[delta, -2.5e+16], N[(lambda1 + t$95$1), $MachinePrecision], If[LessEqual[delta, 3.15e-40], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$1 / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -2.5e16

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   if -2.5e16 < delta < 3.1500000000000001e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 80.9%

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in delta around 0

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

   9. Applied rewrites 77.6%

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

   if 3.1500000000000001e-40 < delta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   9. Applied rewrites 87.0%

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

Alternative 9: 89.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{if}\;delta \leq -2.5 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + t\_1\\ \mathbf{elif}\;delta \leq 3.15 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_1}{\lambda_1}\right) \cdot \lambda_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (atan2 (* (sin delta) (sin theta)) (cos delta))))
   (if (<= delta -2.5e+16)
     (+ lambda1 t_1)
     (if (<= delta 3.15e-40)
       (+
        lambda1
        (atan2
         (* delta (* (cos phi1) (sin theta)))
         (- 1.0 (pow (sin phi1) 2.0))))
       (* (+ 1.0 (/ t_1 lambda1)) lambda1)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((sin(delta) * sin(theta)), cos(delta));
	double tmp;
	if (delta <= -2.5e+16) {
		tmp = lambda1 + t_1;
	} else if (delta <= 3.15e-40) {
		tmp = lambda1 + atan2((delta * (cos(phi1) * sin(theta))), (1.0 - pow(sin(phi1), 2.0)));
	} else {
		tmp = (1.0 + (t_1 / lambda1)) * lambda1;
	}
	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 = atan2((sin(delta) * sin(theta)), cos(delta))
    if (delta <= (-2.5d+16)) then
        tmp = lambda1 + t_1
    else if (delta <= 3.15d-40) then
        tmp = lambda1 + atan2((delta * (cos(phi1) * sin(theta))), (1.0d0 - (sin(phi1) ** 2.0d0)))
    else
        tmp = (1.0d0 + (t_1 / lambda1)) * lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2((Math.sin(delta) * Math.sin(theta)), Math.cos(delta));
	double tmp;
	if (delta <= -2.5e+16) {
		tmp = lambda1 + t_1;
	} else if (delta <= 3.15e-40) {
		tmp = lambda1 + Math.atan2((delta * (Math.cos(phi1) * Math.sin(theta))), (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	} else {
		tmp = (1.0 + (t_1 / lambda1)) * lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((math.sin(delta) * math.sin(theta)), math.cos(delta))
	tmp = 0
	if delta <= -2.5e+16:
		tmp = lambda1 + t_1
	elif delta <= 3.15e-40:
		tmp = lambda1 + math.atan2((delta * (math.cos(phi1) * math.sin(theta))), (1.0 - math.pow(math.sin(phi1), 2.0)))
	else:
		tmp = (1.0 + (t_1 / lambda1)) * lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(sin(delta) * sin(theta)), cos(delta))
	tmp = 0.0
	if (delta <= -2.5e+16)
		tmp = Float64(lambda1 + t_1);
	elseif (delta <= 3.15e-40)
		tmp = Float64(lambda1 + atan(Float64(delta * Float64(cos(phi1) * sin(theta))), Float64(1.0 - (sin(phi1) ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(t_1 / lambda1)) * lambda1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((sin(delta) * sin(theta)), cos(delta));
	tmp = 0.0;
	if (delta <= -2.5e+16)
		tmp = lambda1 + t_1;
	elseif (delta <= 3.15e-40)
		tmp = lambda1 + atan2((delta * (cos(phi1) * sin(theta))), (1.0 - (sin(phi1) ^ 2.0)));
	else
		tmp = (1.0 + (t_1 / lambda1)) * lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[delta, -2.5e+16], N[(lambda1 + t$95$1), $MachinePrecision], If[LessEqual[delta, 3.15e-40], N[(lambda1 + N[ArcTan[N[(delta * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$1 / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -2.5e16

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   if -2.5e16 < delta < 3.1500000000000001e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.8%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 80.9%

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 77.6%

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

   9. Applied rewrites 77.6%

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

   if 3.1500000000000001e-40 < delta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   9. Applied rewrites 87.0%

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

Alternative 10: 87.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 + -0.5 \cdot {delta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -8.5e+105)
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (+ 1.0 (* -0.5 (pow delta 2.0)))))
   (+ lambda1 (atan2 (* (sin delta) (sin theta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -8.5e+105) {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0 + (-0.5 * pow(delta, 2.0))));
	} else {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), cos(delta));
	}
	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) :: tmp
    if (phi1 <= (-8.5d+105)) then
        tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0d0 + ((-0.5d0) * (delta ** 2.0d0))))
    else
        tmp = lambda1 + atan2((sin(delta) * sin(theta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -8.5e+105) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -8.5e+105:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -8.5e+105)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -8.5e+105)
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0 + (-0.5 * (delta ^ 2.0))));
	else
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -8.5e+105], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[delta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.4999999999999999e105

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in delta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 80.0%

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

   7. Applied rewrites 80.0%

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

   if -8.4999999999999999e105 < phi1

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

Alternative 11: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 3.14159:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \sin \left(0.5 \cdot \pi - \phi_1\right)\right)}{1 + -0.5 \cdot {delta}^{2}}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<=
      (atan2
       (* (* (sin theta) (sin delta)) (cos phi1))
       (-
        (cos delta)
        (*
         (sin phi1)
         (sin
          (asin
           (+
            (* (sin phi1) (cos delta))
            (* (* (cos phi1) (sin delta)) (cos theta))))))))
      3.14159)
   (+ lambda1 (atan2 (* (sin delta) (sin theta)) (cos delta)))
   (+
    lambda1
    (atan2
     (* theta (* (sin delta) (sin (- (* 0.5 PI) phi1))))
     (+ 1.0 (* -0.5 (pow delta 2.0)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))) <= 3.14159) {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((theta * (sin(delta) * sin(((0.5 * ((double) M_PI)) - phi1)))), (1.0 + (-0.5 * pow(delta, 2.0))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (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)))))))) <= 3.14159) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((theta * (Math.sin(delta) * Math.sin(((0.5 * Math.PI) - phi1)))), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if 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)))))))) <= 3.14159:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((theta * (math.sin(delta) * math.sin(((0.5 * math.pi) - phi1)))), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (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)))))))) <= 3.14159)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * Float64(sin(delta) * sin(Float64(Float64(0.5 * pi) - phi1)))), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))) <= 3.14159)
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), cos(delta));
	else
		tmp = lambda1 + atan2((theta * (sin(delta) * sin(((0.5 * pi) - phi1)))), (1.0 + (-0.5 * (delta ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[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], 3.14159], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[(N[Sin[delta], $MachinePrecision] * N[Sin[N[(N[(0.5 * Pi), $MachinePrecision] - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[delta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 3.1415899999999999

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   if 3.1415899999999999 < (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 99.8%

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

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 87.1%

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in theta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 74.5%

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

   9. Applied rewrites 74.5%

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

   10. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \sin \left(0.5 \cdot \pi - \phi_1\right)\right)}{1 + \color{blue}{\frac{-1}{2} \cdot {delta}^{2}}} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 70.4%

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

   12. Applied rewrites 70.4%

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

Alternative 12: 81.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;delta \leq -1.65 \cdot 10^{+93}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \left(1 + -0.16666666666666666 \cdot {theta}^{2}\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.6 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 + -0.5 \cdot {delta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -1.65e+93)
   (+
    lambda1
    (atan2
     (* (sin delta) (* theta (+ 1.0 (* -0.16666666666666666 (pow theta 2.0)))))
     (cos delta)))
   (if (<= delta 4.6e-40)
     (+
      lambda1
      (atan2 (* (sin delta) (sin theta)) (+ 1.0 (* -0.5 (pow delta 2.0)))))
     (+ lambda1 (atan2 (* (sin delta) theta) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -1.65e+93) {
		tmp = lambda1 + atan2((sin(delta) * (theta * (1.0 + (-0.16666666666666666 * pow(theta, 2.0))))), cos(delta));
	} else if (delta <= 4.6e-40) {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), (1.0 + (-0.5 * pow(delta, 2.0))));
	} else {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	}
	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) :: tmp
    if (delta <= (-1.65d+93)) then
        tmp = lambda1 + atan2((sin(delta) * (theta * (1.0d0 + ((-0.16666666666666666d0) * (theta ** 2.0d0))))), cos(delta))
    else if (delta <= 4.6d-40) then
        tmp = lambda1 + atan2((sin(delta) * sin(theta)), (1.0d0 + ((-0.5d0) * (delta ** 2.0d0))))
    else
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -1.65e+93) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (theta * (1.0 + (-0.16666666666666666 * Math.pow(theta, 2.0))))), Math.cos(delta));
	} else if (delta <= 4.6e-40) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -1.65e+93:
		tmp = lambda1 + math.atan2((math.sin(delta) * (theta * (1.0 + (-0.16666666666666666 * math.pow(theta, 2.0))))), math.cos(delta))
	elif delta <= 4.6e-40:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -1.65e+93)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(theta * Float64(1.0 + Float64(-0.16666666666666666 * (theta ^ 2.0))))), cos(delta)));
	elseif (delta <= 4.6e-40)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -1.65e+93)
		tmp = lambda1 + atan2((sin(delta) * (theta * (1.0 + (-0.16666666666666666 * (theta ^ 2.0))))), cos(delta));
	elseif (delta <= 4.6e-40)
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), (1.0 + (-0.5 * (delta ^ 2.0))));
	else
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -1.65e+93], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[(1.0 + N[(-0.16666666666666666 * N[Power[theta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 4.6e-40], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[delta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.65e93

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in theta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 73.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \left(1 + -0.16666666666666666 \cdot {theta}^{2}\right)\right)}{\cos delta} \]

   10. Applied rewrites 73.2%

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

   if -1.65e93 < delta < 4.6e-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 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in delta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.7%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 + -0.5 \cdot {delta}^{2}} \]

   10. Applied rewrites 77.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 + \color{blue}{-0.5 \cdot {delta}^{2}}} \]

   if 4.6e-40 < delta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in theta around 0

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

   10. Applied rewrites 74.5%

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

Alternative 13: 81.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 + -0.5 \cdot {delta}^{2}}\\ \mathbf{if}\;theta \leq -38000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 14000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (* (sin delta) (sin theta))
           (+ 1.0 (* -0.5 (pow delta 2.0)))))))
   (if (<= theta -38000.0)
     t_1
     (if (<= theta 14000000000.0)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * sin(theta)), (1.0 + (-0.5 * pow(delta, 2.0))));
	double tmp;
	if (theta <= -38000.0) {
		tmp = t_1;
	} else if (theta <= 14000000000.0) {
		tmp = lambda1 + atan2((sin(delta) * 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((sin(delta) * sin(theta)), (1.0d0 + ((-0.5d0) * (delta ** 2.0d0))))
    if (theta <= (-38000.0d0)) then
        tmp = t_1
    else if (theta <= 14000000000.0d0) then
        tmp = lambda1 + atan2((sin(delta) * 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((Math.sin(delta) * Math.sin(theta)), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	double tmp;
	if (theta <= -38000.0) {
		tmp = t_1;
	} else if (theta <= 14000000000.0) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	tmp = 0
	if theta <= -38000.0:
		tmp = t_1
	elif theta <= 14000000000.0:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))))
	tmp = 0.0
	if (theta <= -38000.0)
		tmp = t_1;
	elseif (theta <= 14000000000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * 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((sin(delta) * sin(theta)), (1.0 + (-0.5 * (delta ^ 2.0))));
	tmp = 0.0;
	if (theta <= -38000.0)
		tmp = t_1;
	elseif (theta <= 14000000000.0)
		tmp = lambda1 + atan2((sin(delta) * 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[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[delta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -38000.0], t$95$1, If[LessEqual[theta, 14000000000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -38000 or 1.4e10 < theta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in delta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.7%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 + -0.5 \cdot {delta}^{2}} \]

   10. Applied rewrites 77.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 + \color{blue}{-0.5 \cdot {delta}^{2}}} \]

   if -38000 < theta < 1.4e10

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in theta around 0

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

   10. Applied rewrites 74.5%

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

Alternative 14: 80.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{if}\;theta \leq -130000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 0.000115:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))))
   (if (<= theta -130000000.0)
     t_1
     (if (<= theta 0.000115)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((delta * sin(theta)), cos(delta));
	double tmp;
	if (theta <= -130000000.0) {
		tmp = t_1;
	} else if (theta <= 0.000115) {
		tmp = lambda1 + atan2((sin(delta) * 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((delta * sin(theta)), cos(delta))
    if (theta <= (-130000000.0d0)) then
        tmp = t_1
    else if (theta <= 0.000115d0) then
        tmp = lambda1 + atan2((sin(delta) * 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((delta * Math.sin(theta)), Math.cos(delta));
	double tmp;
	if (theta <= -130000000.0) {
		tmp = t_1;
	} else if (theta <= 0.000115) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	tmp = 0
	if theta <= -130000000.0:
		tmp = t_1
	elif theta <= 0.000115:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)))
	tmp = 0.0
	if (theta <= -130000000.0)
		tmp = t_1;
	elseif (theta <= 0.000115)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * 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((delta * sin(theta)), cos(delta));
	tmp = 0.0;
	if (theta <= -130000000.0)
		tmp = t_1;
	elseif (theta <= 0.000115)
		tmp = lambda1 + atan2((sin(delta) * 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[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -130000000.0], t$95$1, If[LessEqual[theta, 0.000115], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -1.3e8 or 1.15e-4 < theta

   1. Initial program 99.8%

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

   2. 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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in delta around 0

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

   10. Applied rewrites 75.2%

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

   if -1.3e8 < theta < 1.15e-4

   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. lower-cos.f64 89.2%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 87.1%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in theta around 0

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

   10. Applied rewrites 74.5%

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

Alternative 15: 75.2% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $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. lower-cos.f64 89.2%

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

4. Applied rewrites 89.2%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 87.1%

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

7. Applied rewrites 87.1%

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

8. Taylor expanded in delta around 0

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

10. Applied rewrites 75.2%

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

Alternative 16: 70.9% accurate, 104.4× speedup

Math

\[1 \cdot \lambda_1 \]

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return 1.0 * 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 = 1.0d0 * lambda1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in delta around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-sin.f64 81.2%

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

4. Applied rewrites 81.2%

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

6. Applied rewrites 81.2%

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

7. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{1} \cdot \lambda_1 \]
8. Step-by-step derivation

9. Applied rewrites 70.9%

   \[\leadsto \color{blue}{1} \cdot \lambda_1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(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))))))))))