(- (sqrt PI) (* (- 3/2 (* 1/2 z0)) z0))

Percentage Accurate: 99.2% → 100.0%

Time: 1.1s

Alternatives: 5

Speedup: 1.6×

• 75.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

FPCore

(FPCore (z0)
  :precision binary64
  (- (sqrt PI) (* (- 1.5 (* 0.5 z0)) z0)))

C

double code(double z0) {
	return sqrt(((double) M_PI)) - ((1.5 - (0.5 * z0)) * z0);
}

Java

public static double code(double z0) {
	return Math.sqrt(Math.PI) - ((1.5 - (0.5 * z0)) * z0);
}

Python

def code(z0):
	return math.sqrt(math.pi) - ((1.5 - (0.5 * z0)) * z0)

Julia

function code(z0)
	return Float64(sqrt(pi) - Float64(Float64(1.5 - Float64(0.5 * z0)) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(pi) - ((1.5 - (0.5 * z0)) * z0);
end

Wolfram

code[z0_] := N[(N[Sqrt[Pi], $MachinePrecision] - N[(N[(1.5 - N[(0.5 * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

FPCore

(FPCore (z0)
  :precision binary64
  (- (sqrt PI) (* (- 1.5 (* 0.5 z0)) z0)))

C

double code(double z0) {
	return sqrt(((double) M_PI)) - ((1.5 - (0.5 * z0)) * z0);
}

Java

public static double code(double z0) {
	return Math.sqrt(Math.PI) - ((1.5 - (0.5 * z0)) * z0);
}

Python

def code(z0):
	return math.sqrt(math.pi) - ((1.5 - (0.5 * z0)) * z0)

Julia

function code(z0)
	return Float64(sqrt(pi) - Float64(Float64(1.5 - Float64(0.5 * z0)) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(pi) - ((1.5 - (0.5 * z0)) * z0);
end

Wolfram

code[z0_] := N[(N[Sqrt[Pi], $MachinePrecision] - N[(N[(1.5 - N[(0.5 * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[1.772453850905516 - \left(\left(-0.5 \cdot z0\right) \cdot z0 - -1.5 \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (- 1.772453850905516 (- (* (* -0.5 z0) z0) (* -1.5 z0))))

C

double code(double z0) {
	return 1.772453850905516 - (((-0.5 * z0) * z0) - (-1.5 * 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 = 1.772453850905516d0 - ((((-0.5d0) * z0) * z0) - ((-1.5d0) * z0))
end function

Java

public static double code(double z0) {
	return 1.772453850905516 - (((-0.5 * z0) * z0) - (-1.5 * z0));
}

Python

def code(z0):
	return 1.772453850905516 - (((-0.5 * z0) * z0) - (-1.5 * z0))

Julia

function code(z0)
	return Float64(1.772453850905516 - Float64(Float64(Float64(-0.5 * z0) * z0) - Float64(-1.5 * z0)))
end

MATLAB

function tmp = code(z0)
	tmp = 1.772453850905516 - (((-0.5 * z0) * z0) - (-1.5 * z0));
end

Wolfram

code[z0_] := N[(1.772453850905516 - N[(N[(N[(-0.5 * z0), $MachinePrecision] * z0), $MachinePrecision] - N[(-1.5 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

2. Evaluated real constant 100.0%

   \[\leadsto \color{blue}{1.772453850905516} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\frac{3}{2} - \frac{1}{2} \cdot z0\right) \cdot z0} \]

   2. *-commutative N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{z0 \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot z0\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \color{blue}{\left(\frac{3}{2} - \frac{1}{2} \cdot z0\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \left(\frac{3}{2} - \color{blue}{\frac{1}{2} \cdot z0}\right) \]

   5. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \color{blue}{\left(\frac{3}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z0\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \left(\frac{3}{2} + \color{blue}{\frac{-1}{2}} \cdot z0\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \left(\frac{3}{2} + \color{blue}{\frac{-1}{2} \cdot z0}\right) \]

   8. distribute-rgt-out N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\frac{3}{2} \cdot z0 + \left(\frac{-1}{2} \cdot z0\right) \cdot z0\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\color{blue}{z0 \cdot \frac{3}{2}} + \left(\frac{-1}{2} \cdot z0\right) \cdot z0\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\color{blue}{z0 \cdot \frac{3}{2}} + \left(\frac{-1}{2} \cdot z0\right) \cdot z0\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(z0 \cdot \frac{3}{2} + \color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot z0}\right) \]

   12. +-commutative N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 + z0 \cdot \frac{3}{2}\right)} \]

   13. add-flip N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(z0 \cdot \frac{3}{2}\right)\right)\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(z0 \cdot \frac{3}{2}\right)\right)\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(\color{blue}{z0 \cdot \frac{3}{2}}\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot z0}\right)\right)\right) \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) \cdot z0}\right) \]

   18. metadata-eval N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \color{blue}{\frac{-3}{2}} \cdot z0\right) \]

   19. lower-*.f64 100.0%

       \[\leadsto 1.772453850905516 - \left(\left(-0.5 \cdot z0\right) \cdot z0 - \color{blue}{-1.5 \cdot z0}\right) \]

4. Applied rewrites 100.0%

   \[\leadsto 1.772453850905516 - \color{blue}{\left(\left(-0.5 \cdot z0\right) \cdot z0 - -1.5 \cdot z0\right)} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.6× speedup

Math

\[1.772453850905516 - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

FPCore

(FPCore (z0)
  :precision binary64
  (- 1.772453850905516 (* (- 1.5 (* 0.5 z0)) z0)))

C

double code(double z0) {
	return 1.772453850905516 - ((1.5 - (0.5 * z0)) * 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 = 1.772453850905516d0 - ((1.5d0 - (0.5d0 * z0)) * z0)
end function

Java

public static double code(double z0) {
	return 1.772453850905516 - ((1.5 - (0.5 * z0)) * z0);
}

Python

def code(z0):
	return 1.772453850905516 - ((1.5 - (0.5 * z0)) * z0)

Julia

function code(z0)
	return Float64(1.772453850905516 - Float64(Float64(1.5 - Float64(0.5 * z0)) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = 1.772453850905516 - ((1.5 - (0.5 * z0)) * z0);
end

Wolfram

code[z0_] := N[(1.772453850905516 - N[(N[(1.5 - N[(0.5 * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

2. Evaluated real constant 100.0%

   \[\leadsto \color{blue}{1.772453850905516} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]
3. Add Preprocessing

Alternative 3: 97.9% accurate, 1.9× speedup

Math

\[1.772453850905516 - \left(-0.5 \cdot z0\right) \cdot z0 \]

FPCore

(FPCore (z0)
  :precision binary64
  (- 1.772453850905516 (* (* -0.5 z0) z0)))

C

double code(double z0) {
	return 1.772453850905516 - ((-0.5 * z0) * 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 = 1.772453850905516d0 - (((-0.5d0) * z0) * z0)
end function

Java

public static double code(double z0) {
	return 1.772453850905516 - ((-0.5 * z0) * z0);
}

Python

def code(z0):
	return 1.772453850905516 - ((-0.5 * z0) * z0)

Julia

function code(z0)
	return Float64(1.772453850905516 - Float64(Float64(-0.5 * z0) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = 1.772453850905516 - ((-0.5 * z0) * z0);
end

Wolfram

code[z0_] := N[(1.772453850905516 - N[(N[(-0.5 * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

2. Evaluated real constant 100.0%

   \[\leadsto \color{blue}{1.772453850905516} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\frac{3}{2} - \frac{1}{2} \cdot z0\right) \cdot z0} \]

   2. *-commutative N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{z0 \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot z0\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \color{blue}{\left(\frac{3}{2} - \frac{1}{2} \cdot z0\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \left(\frac{3}{2} - \color{blue}{\frac{1}{2} \cdot z0}\right) \]

   5. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \color{blue}{\left(\frac{3}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z0\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \left(\frac{3}{2} + \color{blue}{\frac{-1}{2}} \cdot z0\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - z0 \cdot \left(\frac{3}{2} + \color{blue}{\frac{-1}{2} \cdot z0}\right) \]

   8. distribute-rgt-out N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\frac{3}{2} \cdot z0 + \left(\frac{-1}{2} \cdot z0\right) \cdot z0\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\color{blue}{z0 \cdot \frac{3}{2}} + \left(\frac{-1}{2} \cdot z0\right) \cdot z0\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\color{blue}{z0 \cdot \frac{3}{2}} + \left(\frac{-1}{2} \cdot z0\right) \cdot z0\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(z0 \cdot \frac{3}{2} + \color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot z0}\right) \]

   12. +-commutative N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 + z0 \cdot \frac{3}{2}\right)} \]

   13. add-flip N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(z0 \cdot \frac{3}{2}\right)\right)\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(z0 \cdot \frac{3}{2}\right)\right)\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(\color{blue}{z0 \cdot \frac{3}{2}}\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot z0}\right)\right)\right) \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) \cdot z0}\right) \]

   18. metadata-eval N/A

       \[\leadsto \frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \color{blue}{\frac{-3}{2}} \cdot z0\right) \]

   19. lower-*.f64 100.0%

       \[\leadsto 1.772453850905516 - \left(\left(-0.5 \cdot z0\right) \cdot z0 - \color{blue}{-1.5 \cdot z0}\right) \]

4. Applied rewrites 100.0%

   \[\leadsto 1.772453850905516 - \color{blue}{\left(\left(-0.5 \cdot z0\right) \cdot z0 - -1.5 \cdot z0\right)} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{7982422502469483}{4503599627370496} - \left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \frac{-3}{2} \cdot z0\right)} \]

   2. sub-negate-rev N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \frac{-3}{2} \cdot z0\right) - \frac{7982422502469483}{4503599627370496}\right)\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \frac{-3}{2} \cdot z0\right)} - \frac{7982422502469483}{4503599627370496}\right)\right) \]

   4. associate--l- N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - \left(\frac{-3}{2} \cdot z0 + \frac{7982422502469483}{4503599627370496}\right)\right)}\right) \]

   5. sub-negate N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{2} \cdot z0 + \frac{7982422502469483}{4503599627370496}\right) - \left(\frac{-1}{2} \cdot z0\right) \cdot z0} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{2} \cdot z0 + \frac{7982422502469483}{4503599627370496}\right) - \left(\frac{-1}{2} \cdot z0\right) \cdot z0} \]

   7. add-flip N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{2} \cdot z0 - \left(\mathsf{neg}\left(\frac{7982422502469483}{4503599627370496}\right)\right)\right)} - \left(\frac{-1}{2} \cdot z0\right) \cdot z0 \]

   8. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{2} \cdot z0 - \left(\mathsf{neg}\left(\frac{7982422502469483}{4503599627370496}\right)\right)\right)} - \left(\frac{-1}{2} \cdot z0\right) \cdot z0 \]

   9. metadata-eval 100.0%

      \[\leadsto \left(-1.5 \cdot z0 - \color{blue}{-1.772453850905516}\right) - \left(-0.5 \cdot z0\right) \cdot z0 \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(-1.5 \cdot z0 - -1.772453850905516\right) - \left(-0.5 \cdot z0\right) \cdot z0} \]

7. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{\frac{7982422502469483}{4503599627370496}} - \left(-0.5 \cdot z0\right) \cdot z0 \]
8. Step-by-step derivation

9. Applied rewrites 97.9%

   \[\leadsto \color{blue}{1.772453850905516} - \left(-0.5 \cdot z0\right) \cdot z0 \]
10. Add Preprocessing

Alternative 4: 52.1% accurate, 3.0× speedup

Math

\[1.772453850905516 - 1.5 \cdot z0 \]

FPCore

(FPCore (z0)
  :precision binary64
  (- 1.772453850905516 (* 1.5 z0)))

C

double code(double z0) {
	return 1.772453850905516 - (1.5 * 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 = 1.772453850905516d0 - (1.5d0 * z0)
end function

Java

public static double code(double z0) {
	return 1.772453850905516 - (1.5 * z0);
}

Python

def code(z0):
	return 1.772453850905516 - (1.5 * z0)

Julia

function code(z0)
	return Float64(1.772453850905516 - Float64(1.5 * z0))
end

MATLAB

function tmp = code(z0)
	tmp = 1.772453850905516 - (1.5 * z0);
end

Wolfram

code[z0_] := N[(1.772453850905516 - N[(1.5 * z0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

2. Evaluated real constant 100.0%

   \[\leadsto \color{blue}{1.772453850905516} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

3. Taylor expanded in z0 around 0

   \[\leadsto 1.772453850905516 - \color{blue}{\frac{3}{2}} \cdot z0 \]
4. Step-by-step derivation

5. Applied rewrites 52.1%

   \[\leadsto 1.772453850905516 - \color{blue}{1.5} \cdot z0 \]
6. Add Preprocessing

Alternative 5: 51.8% accurate, 27.0× speedup

Math

\[1.772453850905516 \]

FPCore

(FPCore (z0)
  :precision binary64
  1.772453850905516)

C

double code(double z0) {
	return 1.772453850905516;
}

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 = 1.772453850905516d0
end function

Java

public static double code(double z0) {
	return 1.772453850905516;
}

Python

def code(z0):
	return 1.772453850905516

Julia

function code(z0)
	return 1.772453850905516
end

MATLAB

function tmp = code(z0)
	tmp = 1.772453850905516;
end

Wolfram

code[z0_] := 1.772453850905516

Derivation

1. Initial program 99.2%

   \[\sqrt{\pi} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

2. Evaluated real constant 100.0%

   \[\leadsto \color{blue}{1.772453850905516} - \left(1.5 - 0.5 \cdot z0\right) \cdot z0 \]

3. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{\frac{7982422502469483}{4503599627370496}} \]
4. Step-by-step derivation

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{1.772453850905516} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(- (sqrt PI) (* (- 3/2 (* 1/2 z0)) z0))"
  :precision binary64
  (- (sqrt PI) (* (- 1.5 (* 0.5 z0)) z0)))