math.square on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%
Time: 991.0ms
Alternatives: 2
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ re \cdot im + im \cdot re \end{array} \]
(FPCore im_sqr (re im) :precision binary64 (+ (* re im) (* im re)))
double im_sqr(double re, double im) {
	return (re * im) + (im * re);
}

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 im_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    im_sqr = (re * im) + (im * re)
end function

public static double im_sqr(double re, double im) {
	return (re * im) + (im * re);
}

def im_sqr(re, im):
	return (re * im) + (im * re)

function im_sqr(re, im)
	return Float64(Float64(re * im) + Float64(im * re))
end

function tmp = im_sqr(re, im)
	tmp = (re * im) + (im * re);
end

im$95$sqr[re_, im_] := N[(N[(re * im), $MachinePrecision] + N[(im * re), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
re \cdot im + im \cdot re
\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ re \cdot im + im \cdot re \end{array} \]
(FPCore im_sqr (re im) :precision binary64 (+ (* re im) (* im re)))
double im_sqr(double re, double im) {
	return (re * im) + (im * re);
}

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 im_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    im_sqr = (re * im) + (im * re)
end function

public static double im_sqr(double re, double im) {
	return (re * im) + (im * re);
}

def im_sqr(re, im):
	return (re * im) + (im * re)

function im_sqr(re, im)
	return Float64(Float64(re * im) + Float64(im * re))
end

function tmp = im_sqr(re, im)
	tmp = (re * im) + (im * re);
end

im$95$sqr[re_, im_] := N[(N[(re * im), $MachinePrecision] + N[(im * re), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
re \cdot im + im \cdot re
\end{array}

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ re \cdot im + im \cdot re \end{array} \]
(FPCore im_sqr (re im) :precision binary64 (+ (* re im) (* im re)))
double im_sqr(double re, double im) {
	return (re * im) + (im * re);
}

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 im_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    im_sqr = (re * im) + (im * re)
end function

public static double im_sqr(double re, double im) {
	return (re * im) + (im * re);
}

def im_sqr(re, im):
	return (re * im) + (im * re)

function im_sqr(re, im)
	return Float64(Float64(re * im) + Float64(im * re))
end

function tmp = im_sqr(re, im)
	tmp = (re * im) + (im * re);
end

im$95$sqr[re_, im_] := N[(N[(re * im), $MachinePrecision] + N[(im * re), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
re \cdot im + im \cdot re
\end{array}
Derivation

  1. Initial program 100.0%

    \[re \cdot im + im \cdot re \]
  2. Add Preprocessing

Alternative 2: 3.3% accurate, 7.0× speedup?

\[\begin{array}{l} \\ 2 \end{array} \]
(FPCore im_sqr (re im) :precision binary64 2.0)
double im_sqr(double re, double im) {
	return 2.0;
}

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 im_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    im_sqr = 2.0d0
end function

public static double im_sqr(double re, double im) {
	return 2.0;
}

def im_sqr(re, im):
	return 2.0

function im_sqr(re, im)
	return 2.0
end

function tmp = im_sqr(re, im)
	tmp = 2.0;
end

im$95$sqr[re_, im_] := 2.0

\begin{array}{l}

\\
2
\end{array}
Derivation

  1. Initial program 100.0%

    \[re \cdot im + im \cdot re \]

  2. Taylor expanded in re around 0

    \[\leadsto \color{blue}{2 \cdot \left(im \cdot re\right)} \]
  3. Step-by-step derivation

    1. count-2-revN/A

      \[\leadsto im \cdot re + \color{blue}{im \cdot re} \]

    2. flip-+N/A

      \[\leadsto \frac{\left(im \cdot re\right) \cdot \left(im \cdot re\right) - \left(im \cdot re\right) \cdot \left(im \cdot re\right)}{\color{blue}{im \cdot re - im \cdot re}} \]

    3. +-inversesN/A

      \[\leadsto \frac{0}{\color{blue}{im \cdot re} - im \cdot re} \]

    4. metadata-evalN/A

      \[\leadsto \frac{1 - 1}{\color{blue}{im \cdot re} - im \cdot re} \]

    5. metadata-evalN/A

      \[\leadsto \frac{1 \cdot 1 - 1}{\color{blue}{im} \cdot re - im \cdot re} \]

    6. metadata-evalN/A

      \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{im \cdot \color{blue}{re} - im \cdot re} \]

    7. +-inversesN/A

      \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} \]

    8. metadata-evalN/A

      \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{1 - \color{blue}{1}} \]

    9. flip-+N/A

      \[\leadsto 1 + \color{blue}{1} \]

    10. metadata-eval3.3

      \[\leadsto 2 \]

  4. Applied rewrites3.3%

    \[\leadsto \color{blue}{2} \]
  5. Add Preprocessing

Reproduce

?
herbie shell --seed 2025101 
(FPCore im_sqr (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))