(cos (* -314159265359/50000000000 z0))

Percentage Accurate: 57.3% → 57.3%

Time: 1.9s

Alternatives: 6

Speedup: 0.6×

Specification

Math

\[\cos \left(-6.28318530718 \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* -6.28318530718 z0)))

C

double code(double z0) {
	return cos((-6.28318530718 * z0));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = cos(((-6.28318530718d0) * z0))
end function

Java

public static double code(double z0) {
	return Math.cos((-6.28318530718 * z0));
}

Python

def code(z0):
	return math.cos((-6.28318530718 * z0))

Julia

function code(z0)
	return cos(Float64(-6.28318530718 * z0))
end

MATLAB

function tmp = code(z0)
	tmp = cos((-6.28318530718 * z0));
end

Wolfram

code[z0_] := N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision]

Initial Program: 57.3% accurate, 1.0× speedup

Math

\[\cos \left(-6.28318530718 \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* -6.28318530718 z0)))

C

double code(double z0) {
	return cos((-6.28318530718 * z0));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = cos(((-6.28318530718d0) * z0))
end function

Java

public static double code(double z0) {
	return Math.cos((-6.28318530718 * z0));
}

Python

def code(z0):
	return math.cos((-6.28318530718 * z0))

Julia

function code(z0)
	return cos(Float64(-6.28318530718 * z0))
end

MATLAB

function tmp = code(z0)
	tmp = cos((-6.28318530718 * z0));
end

Wolfram

code[z0_] := N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision]

Alternative 1: 57.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\left|z0\right|}}\\ \cos \left(\frac{1}{\left(-0.15915494309188485 \cdot t\_0\right) \cdot t\_0}\right) \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (sqrt (/ 1.0 (fabs z0)))))
  (cos (/ 1.0 (* (* -0.15915494309188485 t_0) t_0)))))

C

double code(double z0) {
	double t_0 = sqrt((1.0 / fabs(z0)));
	return cos((1.0 / ((-0.15915494309188485 * t_0) * t_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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: t_0
    t_0 = sqrt((1.0d0 / abs(z0)))
    code = cos((1.0d0 / (((-0.15915494309188485d0) * t_0) * t_0)))
end function

Java

public static double code(double z0) {
	double t_0 = Math.sqrt((1.0 / Math.abs(z0)));
	return Math.cos((1.0 / ((-0.15915494309188485 * t_0) * t_0)));
}

Python

def code(z0):
	t_0 = math.sqrt((1.0 / math.fabs(z0)))
	return math.cos((1.0 / ((-0.15915494309188485 * t_0) * t_0)))

Julia

function code(z0)
	t_0 = sqrt(Float64(1.0 / abs(z0)))
	return cos(Float64(1.0 / Float64(Float64(-0.15915494309188485 * t_0) * t_0)))
end

MATLAB

function tmp = code(z0)
	t_0 = sqrt((1.0 / abs(z0)));
	tmp = cos((1.0 / ((-0.15915494309188485 * t_0) * t_0)));
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Cos[N[(1.0 / N[(N[(-0.15915494309188485 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 57.3%

   \[\cos \left(-6.28318530718 \cdot z0\right) \]
2. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 1\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

   3. associate-/l* N/A

      \[\leadsto \cos \color{blue}{\left(\frac{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}{2}\right)} \]

   4. div-flip N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

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

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot 2}}\right) \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)} \cdot 2}}\right) \]

   9. associate-*l* N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{z0 \cdot \left(\frac{-314159265359}{50000000000} \cdot 2\right)}}}\right) \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   12. metadata-eval 57.3%

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{-12.56637061436} \cdot z0}}\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{-12.56637061436 \cdot z0}}\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   3. associate-/r* N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{\frac{2}{\frac{-314159265359}{25000000000}}}{z0}}}\right) \]

   4. mult-flip N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359}} \cdot \frac{1}{z0}}\right) \]

   7. lower-/.f64 57.2%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\frac{1}{z0}}}\right) \]

5. Applied rewrites 57.2%

   \[\leadsto \cos \left(\frac{1}{\color{blue}{-0.15915494309188485 \cdot \frac{1}{z0}}}\right) \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   2. inv-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{{z0}^{-1}}}\right) \]

   3. sqr-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left(\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   7. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}\right)}\right) \]

   8. lower-unsound-/.f64 28.3%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)}\right) \]

7. Applied rewrites 28.3%

   \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

   3. sqr-abs-rev N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left(\left|{z0}^{\left(\frac{-1}{2}\right)}\right| \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|\right)}}\right) \]

   4. associate-*r* N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\left(\frac{-50000000000}{314159265359} \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\left(\frac{-50000000000}{314159265359} \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\left(\frac{-50000000000}{314159265359} \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|\right)} \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \color{blue}{\sqrt{{z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{{z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   10. pow-sqr N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\color{blue}{{z0}^{\left(2 \cdot \frac{-1}{2}\right)}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{{z0}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   12. metadata-eval N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{{z0}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   13. metadata-eval N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{{z0}^{\color{blue}{-1}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   14. inv-pow N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\color{blue}{\frac{1}{z0}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   15. lift-/.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\color{blue}{\frac{1}{z0}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   16. lower-sqrt.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \color{blue}{\sqrt{\frac{1}{z0}}}\right) \cdot \left|{z0}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

   17. rem-sqrt-square-rev N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\frac{1}{z0}}\right) \cdot \color{blue}{\sqrt{{z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}}}}\right) \]

   18. lift-pow.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\frac{1}{z0}}\right) \cdot \sqrt{\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}}}\right) \]

   19. lift-pow.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\frac{1}{z0}}\right) \cdot \sqrt{{z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}}}\right) \]

   20. pow-sqr N/A

       \[\leadsto \cos \left(\frac{1}{\left(\frac{-50000000000}{314159265359} \cdot \sqrt{\frac{1}{z0}}\right) \cdot \sqrt{\color{blue}{{z0}^{\left(2 \cdot \frac{-1}{2}\right)}}}}\right) \]

9. Applied rewrites 28.3%

   \[\leadsto \cos \left(\frac{1}{\color{blue}{\left(-0.15915494309188485 \cdot \sqrt{\frac{1}{z0}}\right) \cdot \sqrt{\frac{1}{z0}}}}\right) \]
10. Add Preprocessing

Alternative 2: 57.3% accurate, 0.8× speedup

Math

\[\cos \left(\frac{1}{-0.15915494309188485 \cdot \frac{1}{z0}}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (/ 1.0 (* -0.15915494309188485 (/ 1.0 z0)))))

C

double code(double z0) {
	return cos((1.0 / (-0.15915494309188485 * (1.0 / z0))));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = cos((1.0d0 / ((-0.15915494309188485d0) * (1.0d0 / z0))))
end function

Java

public static double code(double z0) {
	return Math.cos((1.0 / (-0.15915494309188485 * (1.0 / z0))));
}

Python

def code(z0):
	return math.cos((1.0 / (-0.15915494309188485 * (1.0 / z0))))

Julia

function code(z0)
	return cos(Float64(1.0 / Float64(-0.15915494309188485 * Float64(1.0 / z0))))
end

MATLAB

function tmp = code(z0)
	tmp = cos((1.0 / (-0.15915494309188485 * (1.0 / z0))));
end

Wolfram

code[z0_] := N[Cos[N[(1.0 / N[(-0.15915494309188485 * N[(1.0 / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.3%

   \[\cos \left(-6.28318530718 \cdot z0\right) \]
2. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 1\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

   3. associate-/l* N/A

      \[\leadsto \cos \color{blue}{\left(\frac{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}{2}\right)} \]

   4. div-flip N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

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

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot 2}}\right) \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)} \cdot 2}}\right) \]

   9. associate-*l* N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{z0 \cdot \left(\frac{-314159265359}{50000000000} \cdot 2\right)}}}\right) \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   12. metadata-eval 57.3%

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{-12.56637061436} \cdot z0}}\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{-12.56637061436 \cdot z0}}\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   3. associate-/r* N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{\frac{2}{\frac{-314159265359}{25000000000}}}{z0}}}\right) \]

   4. mult-flip N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359}} \cdot \frac{1}{z0}}\right) \]

   7. lower-/.f64 57.2%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\frac{1}{z0}}}\right) \]

5. Applied rewrites 57.2%

   \[\leadsto \cos \left(\frac{1}{\color{blue}{-0.15915494309188485 \cdot \frac{1}{z0}}}\right) \]
6. Add Preprocessing

Alternative 3: 57.3% accurate, 0.9× speedup

Math

\[\cos \left(\frac{6.28318530718}{\frac{-1}{z0}}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (/ 6.28318530718 (/ -1.0 z0))))

C

double code(double z0) {
	return cos((6.28318530718 / (-1.0 / z0)));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = cos((6.28318530718d0 / ((-1.0d0) / z0)))
end function

Java

public static double code(double z0) {
	return Math.cos((6.28318530718 / (-1.0 / z0)));
}

Python

def code(z0):
	return math.cos((6.28318530718 / (-1.0 / z0)))

Julia

function code(z0)
	return cos(Float64(6.28318530718 / Float64(-1.0 / z0)))
end

MATLAB

function tmp = code(z0)
	tmp = cos((6.28318530718 / (-1.0 / z0)));
end

Wolfram

code[z0_] := N[Cos[N[(6.28318530718 / N[(-1.0 / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.3%

   \[\cos \left(-6.28318530718 \cdot z0\right) \]
2. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 1\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

   3. associate-/l* N/A

      \[\leadsto \cos \color{blue}{\left(\frac{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}{2}\right)} \]

   4. div-flip N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

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

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot 2}}\right) \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)} \cdot 2}}\right) \]

   9. associate-*l* N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{z0 \cdot \left(\frac{-314159265359}{50000000000} \cdot 2\right)}}}\right) \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   12. metadata-eval 57.3%

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{-12.56637061436} \cdot z0}}\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{-12.56637061436 \cdot z0}}\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   3. associate-/r* N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{\frac{2}{\frac{-314159265359}{25000000000}}}{z0}}}\right) \]

   4. mult-flip N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359}} \cdot \frac{1}{z0}}\right) \]

   7. lower-/.f64 57.2%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\frac{1}{z0}}}\right) \]

5. Applied rewrites 57.2%

   \[\leadsto \cos \left(\frac{1}{\color{blue}{-0.15915494309188485 \cdot \frac{1}{z0}}}\right) \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   2. inv-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{{z0}^{-1}}}\right) \]

   3. sqr-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left(\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   7. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}\right)}\right) \]

   8. lower-unsound-/.f64 28.3%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)}\right) \]

7. Applied rewrites 28.3%

   \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   4. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left(\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)}\right) \]

   6. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}\right)}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

   8. sqr-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{{z0}^{-1}}}\right) \]

   9. inv-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   11. associate-/r* N/A

       \[\leadsto \cos \color{blue}{\left(\frac{\frac{1}{\frac{-50000000000}{314159265359}}}{\frac{1}{z0}}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \cos \left(\frac{\color{blue}{\frac{-314159265359}{50000000000}}}{\frac{1}{z0}}\right) \]

   13. frac-2neg N/A

       \[\leadsto \cos \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)}{\mathsf{neg}\left(\frac{1}{z0}\right)}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \cos \left(\frac{\color{blue}{\frac{314159265359}{50000000000}}}{\mathsf{neg}\left(\frac{1}{z0}\right)}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \cos \color{blue}{\left(\frac{\frac{314159265359}{50000000000}}{\mathsf{neg}\left(\frac{1}{z0}\right)}\right)} \]

   16. lift-/.f64 N/A

       \[\leadsto \cos \left(\frac{\frac{314159265359}{50000000000}}{\mathsf{neg}\left(\color{blue}{\frac{1}{z0}}\right)}\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \cos \left(\frac{\frac{314159265359}{50000000000}}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{z0}}}\right) \]

   18. metadata-eval N/A

       \[\leadsto \cos \left(\frac{\frac{314159265359}{50000000000}}{\frac{\color{blue}{-1}}{z0}}\right) \]

   19. lower-/.f64 57.3%

       \[\leadsto \cos \left(\frac{6.28318530718}{\color{blue}{\frac{-1}{z0}}}\right) \]

9. Applied rewrites 57.3%

   \[\leadsto \cos \color{blue}{\left(\frac{6.28318530718}{\frac{-1}{z0}}\right)} \]
10. Add Preprocessing

Alternative 4: 57.2% accurate, 0.9× speedup

Math

\[\cos \left(\frac{z0}{-0.15915494309188485}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (/ z0 -0.15915494309188485)))

C

double code(double z0) {
	return cos((z0 / -0.15915494309188485));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = cos((z0 / (-0.15915494309188485d0)))
end function

Java

public static double code(double z0) {
	return Math.cos((z0 / -0.15915494309188485));
}

Python

def code(z0):
	return math.cos((z0 / -0.15915494309188485))

Julia

function code(z0)
	return cos(Float64(z0 / -0.15915494309188485))
end

MATLAB

function tmp = code(z0)
	tmp = cos((z0 / -0.15915494309188485));
end

Wolfram

code[z0_] := N[Cos[N[(z0 / -0.15915494309188485), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.3%

   \[\cos \left(-6.28318530718 \cdot z0\right) \]
2. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 1\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \left(\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

   3. associate-/l* N/A

      \[\leadsto \cos \color{blue}{\left(\frac{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}{2}\right)} \]

   4. div-flip N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

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

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}\right)} \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot 2}}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot 2}}\right) \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)} \cdot 2}}\right) \]

   9. associate-*l* N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{z0 \cdot \left(\frac{-314159265359}{50000000000} \cdot 2\right)}}}\right) \]

   10. *-commutative N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\left(\frac{-314159265359}{50000000000} \cdot 2\right) \cdot z0}}}\right) \]

   12. metadata-eval 57.3%

       \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{-12.56637061436} \cdot z0}}\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{2}{-12.56637061436 \cdot z0}}\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{2}{\color{blue}{\frac{-314159265359}{25000000000} \cdot z0}}}\right) \]

   3. associate-/r* N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{\frac{2}{\frac{-314159265359}{25000000000}}}{z0}}}\right) \]

   4. mult-flip N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{2}{\frac{-314159265359}{25000000000}} \cdot \frac{1}{z0}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359}} \cdot \frac{1}{z0}}\right) \]

   7. lower-/.f64 57.2%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\frac{1}{z0}}}\right) \]

5. Applied rewrites 57.2%

   \[\leadsto \cos \left(\frac{1}{\color{blue}{-0.15915494309188485 \cdot \frac{1}{z0}}}\right) \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   2. inv-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{{z0}^{-1}}}\right) \]

   3. sqr-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left(\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   7. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}\right)}\right) \]

   8. lower-unsound-/.f64 28.3%

      \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)}\right) \]

7. Applied rewrites 28.3%

   \[\leadsto \cos \left(\frac{1}{-0.15915494309188485 \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

   4. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left(\color{blue}{{z0}^{\left(\frac{-1}{2}\right)}} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)}\right) \]

   6. lower-unsound-pow.f64 N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \left({z0}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{z0}^{\left(\frac{-1}{2}\right)}}\right)}\right) \]

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

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\left({z0}^{\left(\frac{-1}{2}\right)} \cdot {z0}^{\left(\frac{-1}{2}\right)}\right)}}\right) \]

   8. sqr-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{{z0}^{-1}}}\right) \]

   9. inv-pow N/A

      \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\frac{-50000000000}{314159265359} \cdot \color{blue}{\frac{1}{z0}}}\right) \]

   11. metadata-eval N/A

       \[\leadsto \cos \left(\frac{\color{blue}{1 \cdot 1}}{\frac{-50000000000}{314159265359} \cdot \frac{1}{z0}}\right) \]

   12. *-commutative N/A

       \[\leadsto \cos \left(\frac{1 \cdot 1}{\color{blue}{\frac{1}{z0} \cdot \frac{-50000000000}{314159265359}}}\right) \]

   13. frac-times N/A

       \[\leadsto \cos \color{blue}{\left(\frac{1}{\frac{1}{z0}} \cdot \frac{1}{\frac{-50000000000}{314159265359}}\right)} \]

   14. lift-/.f64 N/A

       \[\leadsto \cos \left(\frac{1}{\color{blue}{\frac{1}{z0}}} \cdot \frac{1}{\frac{-50000000000}{314159265359}}\right) \]

   15. remove-double-div N/A

       \[\leadsto \cos \left(\color{blue}{z0} \cdot \frac{1}{\frac{-50000000000}{314159265359}}\right) \]

   16. metadata-eval N/A

       \[\leadsto \cos \left(z0 \cdot \color{blue}{\frac{-314159265359}{50000000000}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \cos \left(z0 \cdot \color{blue}{\frac{1}{\frac{-50000000000}{314159265359}}}\right) \]

   18. mult-flip-rev N/A

       \[\leadsto \cos \color{blue}{\left(\frac{z0}{\frac{-50000000000}{314159265359}}\right)} \]

   19. lower-/.f64 57.3%

       \[\leadsto \cos \color{blue}{\left(\frac{z0}{-0.15915494309188485}\right)} \]

9. Applied rewrites 57.3%

   \[\leadsto \cos \color{blue}{\left(\frac{z0}{-0.15915494309188485}\right)} \]
10. Add Preprocessing

Alternative 5: 55.8% accurate, 106.0× speedup

Math

\[1 \]

FPCore

(FPCore (z0)
  :precision binary64
  1.0)

C

double code(double z0) {
	return 1.0;
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = 1.0d0
end function

Java

public static double code(double z0) {
	return 1.0;
}

Python

def code(z0):
	return 1.0

Julia

function code(z0)
	return 1.0
end

MATLAB

function tmp = code(z0)
	tmp = 1.0;
end

Wolfram

code[z0_] := 1.0

Derivation

1. Initial program 57.3%

   \[\cos \left(-6.28318530718 \cdot z0\right) \]

2. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 55.8%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(cos (* -314159265359/50000000000 z0))"
  :precision binary64
  (cos (* -6.28318530718 z0)))