Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 20

Speedup: 0.4×

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, 0.2× speedup

Math

\[\begin{array}{l} t_1 := -\sin \phi_1\\ t_2 := t\_1 \cdot \cos \left(-0.5 \cdot \pi\right)\\ t_3 := \sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\\ t_4 := \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \pi\right)\right), \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \left(\left(\cos \phi_1 + \cos \phi_1\right) \cdot 0.5\right), {\left(\cos \left(0.5 \cdot \pi\right) \cdot t\_1\right)}^{3}\right)}{\mathsf{fma}\left(t\_2, t\_2, t\_3 \cdot t\_3 - t\_2 \cdot t\_3\right)}\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot t\_4}{\left(1 - \frac{\mathsf{fma}\left(\cos theta, t\_4 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- (sin phi1)))
        (t_2 (* t_1 (cos (* -0.5 PI))))
        (t_3 (* (sin (* 0.5 PI)) (cos phi1)))
        (t_4
         (/
          (fma
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 PI)))))
           (*
            (+ 0.5 (* 0.5 (cos (* 2.0 phi1))))
            (* (+ (cos phi1) (cos phi1)) 0.5))
           (pow (* (cos (* 0.5 PI)) t_1) 3.0))
          (fma t_2 t_2 (- (* t_3 t_3) (* t_2 t_3))))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) t_4)
     (*
      (-
       1.0
       (/
        (*
         (fma (cos theta) (* t_4 (sin delta)) (* (sin phi1) (cos delta)))
         (sin phi1))
        (cos delta)))
      (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = -sin(phi1);
	double t_2 = t_1 * cos((-0.5 * ((double) M_PI)));
	double t_3 = sin((0.5 * ((double) M_PI))) * cos(phi1);
	double t_4 = fma((0.5 - (0.5 * cos((2.0 * (0.5 * ((double) M_PI)))))), ((0.5 + (0.5 * cos((2.0 * phi1)))) * ((cos(phi1) + cos(phi1)) * 0.5)), pow((cos((0.5 * ((double) M_PI))) * t_1), 3.0)) / fma(t_2, t_2, ((t_3 * t_3) - (t_2 * t_3)));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * t_4), ((1.0 - ((fma(cos(theta), (t_4 * sin(delta)), (sin(phi1) * cos(delta))) * sin(phi1)) / cos(delta))) * cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(-sin(phi1))
	t_2 = Float64(t_1 * cos(Float64(-0.5 * pi)))
	t_3 = Float64(sin(Float64(0.5 * pi)) * cos(phi1))
	t_4 = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * pi))))), Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * phi1)))) * Float64(Float64(cos(phi1) + cos(phi1)) * 0.5)), (Float64(cos(Float64(0.5 * pi)) * t_1) ^ 3.0)) / fma(t_2, t_2, Float64(Float64(t_3 * t_3) - Float64(t_2 * t_3))))
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * t_4), Float64(Float64(1.0 - Float64(Float64(fma(cos(theta), Float64(t_4 * sin(delta)), Float64(sin(phi1) * cos(delta))) * sin(phi1)) / cos(delta))) * cos(delta))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = (-N[Sin[phi1], $MachinePrecision])}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[(-0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] / N[(N[(1.0 - N[(N[(N[(N[Cos[theta], $MachinePrecision] * N[(t$95$4 * N[Sin[delta], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $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

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

   2. sub-to-mult N/A

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

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

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \color{blue}{\frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}}}{\left(1 - \frac{\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \]

7. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}}{\left(1 - \frac{\mathsf{fma}\left(\cos theta, \color{blue}{\frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}} \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \]

9. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \pi\right)\right), \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \left(\left(\cos \phi_1 + \cos \phi_1\right) \cdot 0.5\right), {\left(\cos \left(0.5 \cdot \pi\right) \cdot \left(-\sin \phi_1\right)\right)}^{3}\right)}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}}{\left(1 - \frac{\mathsf{fma}\left(\cos theta, \frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)} \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \]

11. Applied rewrites 99.7%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := -\sin \phi_1\\ t_2 := t\_1 \cdot \cos \left(-0.5 \cdot \pi\right)\\ t_3 := \sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \frac{{t\_2}^{3} + {t\_3}^{3}}{\mathsf{fma}\left(t\_2, t\_2, t\_3 \cdot t\_3 - t\_2 \cdot t\_3\right)}}{\left(1 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_1 + \cos \phi_1, 0.5, \cos \left(0.5 \cdot \pi\right) \cdot t\_1\right), \cos theta \cdot \sin delta, \cos delta \cdot \sin \phi_1\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- (sin phi1)))
        (t_2 (* t_1 (cos (* -0.5 PI))))
        (t_3 (* (sin (* 0.5 PI)) (cos phi1))))
   (+
    lambda1
    (atan2
     (*
      (* (sin theta) (sin delta))
      (/
       (+ (pow t_2 3.0) (pow t_3 3.0))
       (fma t_2 t_2 (- (* t_3 t_3) (* t_2 t_3)))))
     (*
      (-
       1.0
       (/
        (*
         (fma
          (fma (+ (cos phi1) (cos phi1)) 0.5 (* (cos (* 0.5 PI)) t_1))
          (* (cos theta) (sin delta))
          (* (cos delta) (sin phi1)))
         (sin phi1))
        (cos delta)))
      (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = -sin(phi1);
	double t_2 = t_1 * cos((-0.5 * ((double) M_PI)));
	double t_3 = sin((0.5 * ((double) M_PI))) * cos(phi1);
	return lambda1 + atan2(((sin(theta) * sin(delta)) * ((pow(t_2, 3.0) + pow(t_3, 3.0)) / fma(t_2, t_2, ((t_3 * t_3) - (t_2 * t_3))))), ((1.0 - ((fma(fma((cos(phi1) + cos(phi1)), 0.5, (cos((0.5 * ((double) M_PI))) * t_1)), (cos(theta) * sin(delta)), (cos(delta) * sin(phi1))) * sin(phi1)) / cos(delta))) * cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(-sin(phi1))
	t_2 = Float64(t_1 * cos(Float64(-0.5 * pi)))
	t_3 = Float64(sin(Float64(0.5 * pi)) * cos(phi1))
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * Float64(Float64((t_2 ^ 3.0) + (t_3 ^ 3.0)) / fma(t_2, t_2, Float64(Float64(t_3 * t_3) - Float64(t_2 * t_3))))), Float64(Float64(1.0 - Float64(Float64(fma(fma(Float64(cos(phi1) + cos(phi1)), 0.5, Float64(cos(Float64(0.5 * pi)) * t_1)), Float64(cos(theta) * sin(delta)), Float64(cos(delta) * sin(phi1))) * sin(phi1)) / cos(delta))) * cos(delta))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = (-N[Sin[phi1], $MachinePrecision])}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[(-0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(N[(N[(N[(N[(N[Cos[phi1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $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

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

   2. sub-to-mult N/A

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

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

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \color{blue}{\frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}}}{\left(1 - \frac{\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \]

7. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}}{\left(1 - \frac{\mathsf{fma}\left(\cos theta, \color{blue}{\frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}} \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \]

9. Applied rewrites 99.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \frac{{\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right)}^{3} + {\left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)}^{3}}{\mathsf{fma}\left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right), \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right) - \left(\left(-\sin \phi_1\right) \cdot \cos \left(-0.5 \cdot \pi\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \cos \phi_1\right)\right)}}{\left(1 - \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_1 + \cos \phi_1, 0.5, \cos \left(0.5 \cdot \pi\right) \cdot \left(-\sin \phi_1\right)\right), \cos theta \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)} \cdot \sin \phi_1}{\cos delta}\right) \cdot \cos delta} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta))))
   (+
    lambda1
    (atan2
     (fma
      t_1
      (* (- (sin phi1)) (cos (* -0.5 PI)))
      (* t_1 (* (sin (* 0.5 PI)) (cos phi1))))
     (*
      (-
       1.0
       (/
        (*
         (fma
          (cos theta)
          (* (cos phi1) (sin delta))
          (* (sin phi1) (cos delta)))
         (sin phi1))
        (cos delta)))
      (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	return lambda1 + atan2(fma(t_1, (-sin(phi1) * cos((-0.5 * ((double) M_PI)))), (t_1 * (sin((0.5 * ((double) M_PI))) * cos(phi1)))), ((1.0 - ((fma(cos(theta), (cos(phi1) * sin(delta)), (sin(phi1) * cos(delta))) * sin(phi1)) / cos(delta))) * cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	return Float64(lambda1 + atan(fma(t_1, Float64(Float64(-sin(phi1)) * cos(Float64(-0.5 * pi))), Float64(t_1 * Float64(sin(Float64(0.5 * pi)) * cos(phi1)))), Float64(Float64(1.0 - Float64(Float64(fma(cos(theta), Float64(cos(phi1) * sin(delta)), Float64(sin(phi1) * cos(delta))) * sin(phi1)) / cos(delta))) * cos(delta))))
end

Wolfram

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

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

   2. sub-to-mult N/A

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

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

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (sin delta))))
   (+
    lambda1
    (atan2
     (* t_1 (sin theta))
     (-
      (cos delta)
      (*
       (sin phi1)
       (sin (asin (+ (* (sin phi1) (cos delta)) (* t_1 (cos theta)))))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * sin(delta);
	return lambda1 + atan2((t_1 * sin(theta)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + (t_1 * 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
    real(8) :: t_1
    t_1 = cos(phi1) * sin(delta)
    code = lambda1 + atan2((t_1 * sin(theta)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + (t_1 * cos(theta))))))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * Math.sin(delta);
	return lambda1 + Math.atan2((t_1 * Math.sin(theta)), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.sin(phi1) * Math.cos(delta)) + (t_1 * Math.cos(theta))))))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * math.sin(delta)
	return lambda1 + math.atan2((t_1 * math.sin(theta)), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.sin(phi1) * math.cos(delta)) + (t_1 * math.cos(theta))))))))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * sin(delta);
	tmp = lambda1 + atan2((t_1 * sin(theta)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + (t_1 * cos(theta))))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 99.8

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

3. Applied rewrites 99.8%

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

Alternative 5: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 99.8

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

3. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 6: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.8

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

3. Applied rewrites 99.8%

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

Alternative 7: 94.5% 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 94.5

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

4. Applied rewrites 94.5%

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

Alternative 8: 92.0% accurate, 2.0× 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 92.0

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

4. Applied rewrites 92.0%

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

Alternative 9: 88.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi1 < 8.5

   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 88.5

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

   4. Applied rewrites 88.5%

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

   if 8.5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 80.2

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

   6. Applied rewrites 80.2%

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

Alternative 10: 88.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq 8.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 - {\sin \phi_1}^{2}}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1))))
   (if (<= phi1 8.5)
     (+ lambda1 (atan2 t_1 (cos delta)))
     (+ lambda1 (atan2 t_1 (- 1.0 (pow (sin phi1) 2.0)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double tmp;
	if (phi1 <= 8.5) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, (1.0 - pow(sin(phi1), 2.0)));
	}
	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 = (sin(theta) * sin(delta)) * cos(phi1)
    if (phi1 <= 8.5d0) then
        tmp = lambda1 + atan2(t_1, cos(delta))
    else
        tmp = lambda1 + atan2(t_1, (1.0d0 - (sin(phi1) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1);
	double tmp;
	if (phi1 <= 8.5) {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = (math.sin(theta) * math.sin(delta)) * math.cos(phi1)
	tmp = 0
	if phi1 <= 8.5:
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(t_1, (1.0 - math.pow(math.sin(phi1), 2.0)))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	tmp = 0.0
	if (phi1 <= 8.5)
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 - (sin(phi1) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	tmp = 0.0;
	if (phi1 <= 8.5)
		tmp = lambda1 + atan2(t_1, cos(delta));
	else
		tmp = lambda1 + atan2(t_1, (1.0 - (sin(phi1) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 8.5], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 8.5

   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 88.5

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

   4. Applied rewrites 88.5%

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

   if 8.5 < 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 delta around 0

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

      1. lower--.f64 N/A

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

      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 80.2

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

   4. Applied rewrites 80.2%

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

Alternative 11: 88.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 88.5

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

4. Applied rewrites 88.5%

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

Alternative 12: 87.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot \cos \phi_1}{1 + -0.5 \cdot {delta}^{2}}\\ \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{+50}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 3.6:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta)))
        (t_2
         (+
          lambda1
          (atan2 (* t_1 (cos phi1)) (+ 1.0 (* -0.5 (pow delta 2.0)))))))
   (if (<= phi1 -6.8e+50)
     t_2
     (if (<= phi1 3.6)
       (+ lambda1 (atan2 (* t_1 (+ 1.0 (* -0.5 (pow phi1 2.0)))) (cos delta)))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double t_2 = lambda1 + atan2((t_1 * cos(phi1)), (1.0 + (-0.5 * pow(delta, 2.0))));
	double tmp;
	if (phi1 <= -6.8e+50) {
		tmp = t_2;
	} else if (phi1 <= 3.6) {
		tmp = lambda1 + atan2((t_1 * (1.0 + (-0.5 * pow(phi1, 2.0)))), cos(delta));
	} 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)
    t_2 = lambda1 + atan2((t_1 * cos(phi1)), (1.0d0 + ((-0.5d0) * (delta ** 2.0d0))))
    if (phi1 <= (-6.8d+50)) then
        tmp = t_2
    else if (phi1 <= 3.6d0) then
        tmp = lambda1 + atan2((t_1 * (1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0)))), cos(delta))
    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);
	double t_2 = lambda1 + Math.atan2((t_1 * Math.cos(phi1)), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	double tmp;
	if (phi1 <= -6.8e+50) {
		tmp = t_2;
	} else if (phi1 <= 3.6) {
		tmp = lambda1 + Math.atan2((t_1 * (1.0 + (-0.5 * Math.pow(phi1, 2.0)))), Math.cos(delta));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(theta) * math.sin(delta)
	t_2 = lambda1 + math.atan2((t_1 * math.cos(phi1)), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	tmp = 0
	if phi1 <= -6.8e+50:
		tmp = t_2
	elif phi1 <= 3.6:
		tmp = lambda1 + math.atan2((t_1 * (1.0 + (-0.5 * math.pow(phi1, 2.0)))), math.cos(delta))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	t_2 = Float64(lambda1 + atan(Float64(t_1 * cos(phi1)), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))))
	tmp = 0.0
	if (phi1 <= -6.8e+50)
		tmp = t_2;
	elseif (phi1 <= 3.6)
		tmp = Float64(lambda1 + atan(Float64(t_1 * Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0)))), cos(delta)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(theta) * sin(delta);
	t_2 = lambda1 + atan2((t_1 * cos(phi1)), (1.0 + (-0.5 * (delta ^ 2.0))));
	tmp = 0.0;
	if (phi1 <= -6.8e+50)
		tmp = t_2;
	elseif (phi1 <= 3.6)
		tmp = lambda1 + atan2((t_1 * (1.0 + (-0.5 * (phi1 ^ 2.0)))), cos(delta));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.5 * N[Power[delta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -6.8e+50], t$95$2, If[LessEqual[phi1, 3.6], N[(lambda1 + N[ArcTan[N[(t$95$1 * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.7999999999999997e50 or 3.60000000000000009 < 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 88.5

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

   4. Applied rewrites 88.5%

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

   5. 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 79.5

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

   7. Applied rewrites 79.5%

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

   if -6.7999999999999997e50 < phi1 < 3.60000000000000009

   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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 13: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 3.05:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1 \cdot \sin 1.5707963267948966}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 + -0.5 \cdot {delta}^{2}}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta))))
   (if (<=
        (atan2
         (* t_1 (cos phi1))
         (-
          (cos delta)
          (*
           (sin phi1)
           (sin
            (asin
             (+
              (* (sin phi1) (cos delta))
              (* (* (cos phi1) (sin delta)) (cos theta))))))))
        3.05)
     (+ lambda1 (atan2 (* t_1 (sin 1.5707963267948966)) (cos delta)))
     (+
      lambda1
      (atan2
       (* (* theta (sin delta)) (cos phi1))
       (+ 1.0 (* -0.5 (pow delta 2.0))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double tmp;
	if (atan2((t_1 * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))) <= 3.05) {
		tmp = lambda1 + atan2((t_1 * sin(1.5707963267948966)), cos(delta));
	} else {
		tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), (1.0 + (-0.5 * pow(delta, 2.0))));
	}
	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 = sin(theta) * sin(delta)
    if (atan2((t_1 * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))) <= 3.05d0) then
        tmp = lambda1 + atan2((t_1 * sin(1.5707963267948966d0)), cos(delta))
    else
        tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), (1.0d0 + ((-0.5d0) * (delta ** 2.0d0))))
    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);
	double tmp;
	if (Math.atan2((t_1 * 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)))))))) <= 3.05) {
		tmp = lambda1 + Math.atan2((t_1 * Math.sin(1.5707963267948966)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(((theta * Math.sin(delta)) * Math.cos(phi1)), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(theta) * math.sin(delta)
	tmp = 0
	if math.atan2((t_1 * 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)))))))) <= 3.05:
		tmp = lambda1 + math.atan2((t_1 * math.sin(1.5707963267948966)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(((theta * math.sin(delta)) * math.cos(phi1)), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	tmp = 0.0
	if (atan(Float64(t_1 * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)))))))) <= 3.05)
		tmp = Float64(lambda1 + atan(Float64(t_1 * sin(1.5707963267948966)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(theta * sin(delta)) * cos(phi1)), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(theta) * sin(delta);
	tmp = 0.0;
	if (atan2((t_1 * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))) <= 3.05)
		tmp = lambda1 + atan2((t_1 * sin(1.5707963267948966)), cos(delta));
	else
		tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), (1.0 + (-0.5 * (delta ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcTan[N[(t$95$1 * 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], 3.05], N[(lambda1 + N[ArcTan[N[(t$95$1 * N[Sin[1.5707963267948966], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(theta * 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]]]

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)))))))) < 3.0499999999999998

   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 88.5

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

   4. Applied rewrites 88.5%

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

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

      9. lower-PI.f64 N/A

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

      10. metadata-eval 86.0

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

   6. Applied rewrites 86.0%

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

   7. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 86.2

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

   9. Applied rewrites 86.2%

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

   10. Evaluated real constant 86.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \sin \frac{884279719003555}{562949953421312}}{\cos delta} \]

   if 3.0499999999999998 < (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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in delta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 70.9

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

   10. Applied rewrites 70.9%

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

Alternative 14: 84.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{+84}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta}\\ \mathbf{elif}\;\phi_1 \leq 82000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot delta\right) \cdot \cos \phi_1}{\cos delta}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -2.3e+84)
   (+ lambda1 (atan2 (* (* theta (sin delta)) (cos phi1)) (cos delta)))
   (if (<= phi1 82000000.0)
     (+
      lambda1
      (atan2
       (* (* (sin theta) (sin delta)) (+ 1.0 (* -0.5 (pow phi1 2.0))))
       (cos delta)))
     (+ lambda1 (atan2 (* (* (sin theta) delta) (cos phi1)) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -2.3e+84) {
		tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta));
	} else if (phi1 <= 82000000.0) {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * (1.0 + (-0.5 * pow(phi1, 2.0)))), cos(delta));
	} else {
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), 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 <= (-2.3d+84)) then
        tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta))
    else if (phi1 <= 82000000.0d0) then
        tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * (1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0)))), cos(delta))
    else
        tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), 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 <= -2.3e+84) {
		tmp = lambda1 + Math.atan2(((theta * Math.sin(delta)) * Math.cos(phi1)), Math.cos(delta));
	} else if (phi1 <= 82000000.0) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * (1.0 + (-0.5 * Math.pow(phi1, 2.0)))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * delta) * Math.cos(phi1)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -2.3e+84:
		tmp = lambda1 + math.atan2(((theta * math.sin(delta)) * math.cos(phi1)), math.cos(delta))
	elif phi1 <= 82000000.0:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * (1.0 + (-0.5 * math.pow(phi1, 2.0)))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(((math.sin(theta) * delta) * math.cos(phi1)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -2.3e+84)
		tmp = Float64(lambda1 + atan(Float64(Float64(theta * sin(delta)) * cos(phi1)), cos(delta)));
	elseif (phi1 <= 82000000.0)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0)))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * delta) * cos(phi1)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -2.3e+84)
		tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta));
	elseif (phi1 <= 82000000.0)
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * (1.0 + (-0.5 * (phi1 ^ 2.0)))), cos(delta));
	else
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -2.3e+84], N[(lambda1 + N[ArcTan[N[(N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 82000000.0], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.2999999999999999e84

   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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

   if -2.2999999999999999e84 < phi1 < 8.2e7

   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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 76.9

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

   7. Applied rewrites 76.9%

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

   if 8.2e7 < 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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in delta around 0

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

   7. Applied rewrites 74.6%

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

Alternative 15: 81.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta}\\ \mathbf{if}\;delta \leq -2.8 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 5.5 \cdot 10^{+36}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot delta\right) \cdot \cos \phi_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+ lambda1 (atan2 (* (* theta (sin delta)) (cos phi1)) (cos delta)))))
   (if (<= delta -2.8e+19)
     t_1
     (if (<= delta 5.5e+36)
       (+ lambda1 (atan2 (* (* (sin theta) delta) (cos phi1)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta));
	double tmp;
	if (delta <= -2.8e+19) {
		tmp = t_1;
	} else if (delta <= 5.5e+36) {
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(lambda1, phi1, phi2, delta, theta)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta))
    if (delta <= (-2.8d+19)) then
        tmp = t_1
    else if (delta <= 5.5d+36) then
        tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2(((theta * Math.sin(delta)) * Math.cos(phi1)), Math.cos(delta));
	double tmp;
	if (delta <= -2.8e+19) {
		tmp = t_1;
	} else if (delta <= 5.5e+36) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * delta) * Math.cos(phi1)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2(((theta * math.sin(delta)) * math.cos(phi1)), math.cos(delta))
	tmp = 0
	if delta <= -2.8e+19:
		tmp = t_1
	elif delta <= 5.5e+36:
		tmp = lambda1 + math.atan2(((math.sin(theta) * delta) * math.cos(phi1)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(Float64(theta * sin(delta)) * cos(phi1)), cos(delta)))
	tmp = 0.0
	if (delta <= -2.8e+19)
		tmp = t_1;
	elseif (delta <= 5.5e+36)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * delta) * cos(phi1)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta));
	tmp = 0.0;
	if (delta <= -2.8e+19)
		tmp = t_1;
	elseif (delta <= 5.5e+36)
		tmp = lambda1 + atan2(((sin(theta) * delta) * cos(phi1)), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2.8e+19], t$95$1, If[LessEqual[delta, 5.5e+36], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.8e19 or 5.5000000000000002e36 < 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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

   if -2.8e19 < delta < 5.5000000000000002e36

   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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in delta around 0

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

   7. Applied rewrites 74.6%

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

Alternative 16: 76.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;theta \leq -45000:\\ \;\;\;\;1 \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta -45000.0)
   (* 1.0 lambda1)
   (+ lambda1 (atan2 (* (* theta (sin delta)) (cos phi1)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -45000.0) {
		tmp = 1.0 * lambda1;
	} else {
		tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), 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 (theta <= (-45000.0d0)) then
        tmp = 1.0d0 * lambda1
    else
        tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -45000.0) {
		tmp = 1.0 * lambda1;
	} else {
		tmp = lambda1 + Math.atan2(((theta * Math.sin(delta)) * Math.cos(phi1)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if theta <= -45000.0:
		tmp = 1.0 * lambda1
	else:
		tmp = lambda1 + math.atan2(((theta * math.sin(delta)) * math.cos(phi1)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= -45000.0)
		tmp = Float64(1.0 * lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(theta * sin(delta)) * cos(phi1)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (theta <= -45000.0)
		tmp = 1.0 * lambda1;
	else
		tmp = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, -45000.0], N[(1.0 * lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -45000

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 69.7%

      \[\leadsto \color{blue}{1} \cdot \lambda_1 \]

   if -45000 < 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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

Alternative 17: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 + -0.5 \cdot {delta}^{2}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-138}:\\ \;\;\;\;1 \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (atan2
          (* (* (sin theta) (sin delta)) (cos phi1))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (sin phi1) (cos delta))
               (* (* (cos phi1) (sin delta)) (cos theta)))))))))
        (t_2
         (+
          lambda1
          (atan2
           (* (* theta (sin delta)) (cos phi1))
           (+ 1.0 (* -0.5 (pow delta 2.0)))))))
   (if (<= t_1 -1e-16) t_2 (if (<= t_1 4e-138) (* 1.0 lambda1) t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	double t_2 = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), (1.0 + (-0.5 * pow(delta, 2.0))));
	double tmp;
	if (t_1 <= -1e-16) {
		tmp = t_2;
	} else if (t_1 <= 4e-138) {
		tmp = 1.0 * lambda1;
	} 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 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))
    t_2 = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), (1.0d0 + ((-0.5d0) * (delta ** 2.0d0))))
    if (t_1 <= (-1d-16)) then
        tmp = t_2
    else if (t_1 <= 4d-138) then
        tmp = 1.0d0 * lambda1
    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.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))))))));
	double t_2 = lambda1 + Math.atan2(((theta * Math.sin(delta)) * Math.cos(phi1)), (1.0 + (-0.5 * Math.pow(delta, 2.0))));
	double tmp;
	if (t_1 <= -1e-16) {
		tmp = t_2;
	} else if (t_1 <= 4e-138) {
		tmp = 1.0 * lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = 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))))))))
	t_2 = lambda1 + math.atan2(((theta * math.sin(delta)) * math.cos(phi1)), (1.0 + (-0.5 * math.pow(delta, 2.0))))
	tmp = 0
	if t_1 <= -1e-16:
		tmp = t_2
	elif t_1 <= 4e-138:
		tmp = 1.0 * lambda1
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))))
	t_2 = Float64(lambda1 + atan(Float64(Float64(theta * sin(delta)) * cos(phi1)), Float64(1.0 + Float64(-0.5 * (delta ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-16)
		tmp = t_2;
	elseif (t_1 <= 4e-138)
		tmp = Float64(1.0 * lambda1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	t_2 = lambda1 + atan2(((theta * sin(delta)) * cos(phi1)), (1.0 + (-0.5 * (delta ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= -1e-16)
		tmp = t_2;
	elseif (t_1 <= 4e-138)
		tmp = 1.0 * lambda1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(N[(theta * 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]}, If[LessEqual[t$95$1, -1e-16], t$95$2, If[LessEqual[t$95$1, 4e-138], N[(1.0 * lambda1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -9.9999999999999998e-17 or 4.00000000000000027e-138 < (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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in delta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 70.9

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

   10. Applied rewrites 70.9%

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

   if -9.9999999999999998e-17 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 4.00000000000000027e-138

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 69.7%

      \[\leadsto \color{blue}{1} \cdot \lambda_1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot \sin delta\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)}{\cos delta}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-98}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-102}:\\ \;\;\;\;1 \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (atan2
          (* (* (sin theta) (sin delta)) (cos phi1))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (sin phi1) (cos delta))
               (* (* (cos phi1) (sin delta)) (cos theta)))))))))
        (t_2
         (+
          lambda1
          (atan2
           (* (* theta (sin delta)) (+ 1.0 (* -0.5 (pow phi1 2.0))))
           (cos delta)))))
   (if (<= t_1 -5e-98) t_2 (if (<= t_1 4e-102) (* 1.0 lambda1) t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	double t_2 = lambda1 + atan2(((theta * sin(delta)) * (1.0 + (-0.5 * pow(phi1, 2.0)))), cos(delta));
	double tmp;
	if (t_1 <= -5e-98) {
		tmp = t_2;
	} else if (t_1 <= 4e-102) {
		tmp = 1.0 * lambda1;
	} 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 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))
    t_2 = lambda1 + atan2(((theta * sin(delta)) * (1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0)))), cos(delta))
    if (t_1 <= (-5d-98)) then
        tmp = t_2
    else if (t_1 <= 4d-102) then
        tmp = 1.0d0 * lambda1
    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.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))))))));
	double t_2 = lambda1 + Math.atan2(((theta * Math.sin(delta)) * (1.0 + (-0.5 * Math.pow(phi1, 2.0)))), Math.cos(delta));
	double tmp;
	if (t_1 <= -5e-98) {
		tmp = t_2;
	} else if (t_1 <= 4e-102) {
		tmp = 1.0 * lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = 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))))))))
	t_2 = lambda1 + math.atan2(((theta * math.sin(delta)) * (1.0 + (-0.5 * math.pow(phi1, 2.0)))), math.cos(delta))
	tmp = 0
	if t_1 <= -5e-98:
		tmp = t_2
	elif t_1 <= 4e-102:
		tmp = 1.0 * lambda1
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))))
	t_2 = Float64(lambda1 + atan(Float64(Float64(theta * sin(delta)) * Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0)))), cos(delta)))
	tmp = 0.0
	if (t_1 <= -5e-98)
		tmp = t_2;
	elseif (t_1 <= 4e-102)
		tmp = Float64(1.0 * lambda1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	t_2 = lambda1 + atan2(((theta * sin(delta)) * (1.0 + (-0.5 * (phi1 ^ 2.0)))), cos(delta));
	tmp = 0.0;
	if (t_1 <= -5e-98)
		tmp = t_2;
	elseif (t_1 <= 4e-102)
		tmp = 1.0 * lambda1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-98], t$95$2, If[LessEqual[t$95$1, 4e-102], N[(1.0 * lambda1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -5.00000000000000018e-98 or 3.99999999999999973e-102 < (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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 65.9

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

   10. Applied rewrites 65.9%

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

   if -5.00000000000000018e-98 < (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)))))))) < 3.99999999999999973e-102

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 69.7%

      \[\leadsto \color{blue}{1} \cdot \lambda_1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 69.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;theta \leq -2 \cdot 10^{-277}:\\ \;\;\;\;1 \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(theta \cdot delta\right) \cdot \cos \phi_1}{\cos delta}\\ \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta -2e-277)
   (* 1.0 lambda1)
   (+ lambda1 (atan2 (* (* theta delta) (cos phi1)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -2e-277) {
		tmp = 1.0 * lambda1;
	} else {
		tmp = lambda1 + atan2(((theta * delta) * cos(phi1)), 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 (theta <= (-2d-277)) then
        tmp = 1.0d0 * lambda1
    else
        tmp = lambda1 + atan2(((theta * delta) * cos(phi1)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -2e-277) {
		tmp = 1.0 * lambda1;
	} else {
		tmp = lambda1 + Math.atan2(((theta * delta) * Math.cos(phi1)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if theta <= -2e-277:
		tmp = 1.0 * lambda1
	else:
		tmp = lambda1 + math.atan2(((theta * delta) * math.cos(phi1)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= -2e-277)
		tmp = Float64(1.0 * lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(theta * delta) * cos(phi1)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (theta <= -2e-277)
		tmp = 1.0 * lambda1;
	else
		tmp = lambda1 + atan2(((theta * delta) * cos(phi1)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, -2e-277], N[(1.0 * lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(theta * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -1.99999999999999994e-277

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 69.7%

      \[\leadsto \color{blue}{1} \cdot \lambda_1 \]

   if -1.99999999999999994e-277 < 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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in theta around 0

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in delta around 0

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

   10. Applied rewrites 67.1%

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

Alternative 20: 68.4% accurate, 100.4× 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 phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 76.9

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

4. Applied rewrites 76.9%

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

   1. lift-+.f64 N/A

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

   2. sum-to-mult N/A

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

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

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

6. Applied rewrites 76.9%

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

7. Taylor expanded in lambda1 around inf

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

9. Applied rewrites 69.7%

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

Reproduce

herbie shell --seed 2025178 
(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))))))))))