math.square on complex, real part

Percentage Accurate: 93.7% → 100.0%
Time: 1.3s
Alternatives: 3
Speedup: 1.2×

Specification

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

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

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

def re_sqr(re, im):
	return (re * re) - (im * im)

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

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

\begin{array}{l}

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

Sampling outcomes in binary64 precision:

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 3 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: 93.7% accurate, 1.0× speedup?

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

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

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

def re_sqr(re, im):
	return (re * re) - (im * im)

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

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

\begin{array}{l}

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

Alternative 1: 100.0% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \left(re - im\right) \cdot \left(im + re\right) \end{array} \]
(FPCore re_sqr (re im) :precision binary64 (* (- re im) (+ im re)))
double re_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 re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re - im) * (im + re)
end function

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

def re_sqr(re, im):
	return (re - im) * (im + re)

function re_sqr(re, im)
	return Float64(Float64(re - im) * Float64(im + re))
end

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

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

\begin{array}{l}

\\
\left(re - im\right) \cdot \left(im + re\right)
\end{array}
Derivation

  1. Initial program 93.7%

    \[re \cdot re - im \cdot im \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift--.f64N/A

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

    2. lift-*.f64N/A

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

    3. lift-*.f64N/A

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

    4. difference-of-squaresN/A

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

    5. *-commutativeN/A

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

    6. lower-*.f64N/A

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

    7. lower--.f64N/A

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

    8. +-commutativeN/A

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

    9. lower-+.f64100.0

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

  4. Applied rewrites100.0%

    \[\leadsto \color{blue}{\left(re - im\right) \cdot \left(im + re\right)} \]
  5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq -5 \cdot 10^{-315}:\\ \;\;\;\;\left(-im\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (- (* re re) (* im im)) -5e-315) (* (- im) im) (* re re)))
double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) - (im * im)) <= -5e-315) {
		tmp = -im * im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

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 re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((re * re) - (im * im)) <= (-5d-315)) then
        tmp = -im * im
    else
        tmp = re * re
    end if
    re_sqr = tmp
end function

public static double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) - (im * im)) <= -5e-315) {
		tmp = -im * im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

def re_sqr(re, im):
	tmp = 0
	if ((re * re) - (im * im)) <= -5e-315:
		tmp = -im * im
	else:
		tmp = re * re
	return tmp

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(Float64(re * re) - Float64(im * im)) <= -5e-315)
		tmp = Float64(Float64(-im) * im);
	else
		tmp = Float64(re * re);
	end
	return tmp
end

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (((re * re) - (im * im)) <= -5e-315)
		tmp = -im * im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

re$95$sqr[re_, im_] := If[LessEqual[N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], -5e-315], N[((-im) * im), $MachinePrecision], N[(re * re), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;re \cdot re - im \cdot im \leq -5 \cdot 10^{-315}:\\
\;\;\;\;\left(-im\right) \cdot im\\

\mathbf{else}:\\
\;\;\;\;re \cdot re\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -5.0000000023e-315

    1. Initial program 100.0%

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

    3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
    4. Step-by-step derivation

      1. mul-1-negN/A

        \[\leadsto \mathsf{neg}\left({im}^{2}\right) \]

      2. pow2N/A

        \[\leadsto \mathsf{neg}\left(im \cdot im\right) \]

      3. distribute-lft-neg-inN/A

        \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \color{blue}{im} \]

      4. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \color{blue}{im} \]

      5. lower-neg.f6499.6

        \[\leadsto \left(-im\right) \cdot im \]

    5. Applied rewrites99.6%

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

    if -5.0000000023e-315 < (-.f64 (*.f64 re re) (*.f64 im im))

    1. Initial program 88.4%

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

    3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{2}} \]
    4. Step-by-step derivation

      1. pow2N/A

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

      2. lift-*.f6495.0

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

    5. Applied rewrites95.0%

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

Alternative 3: 52.4% accurate, 2.3× speedup?

\[\begin{array}{l} \\ re \cdot re \end{array} \]
(FPCore re_sqr (re im) :precision binary64 (* re re))
double re_sqr(double re, double im) {
	return re * 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 re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = re * re
end function

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

def re_sqr(re, im):
	return re * re

function re_sqr(re, im)
	return Float64(re * re)
end

function tmp = re_sqr(re, im)
	tmp = re * re;
end

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

\begin{array}{l}

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

  1. Initial program 93.7%

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

  3. Taylor expanded in re around inf

    \[\leadsto \color{blue}{{re}^{2}} \]
  4. Step-by-step derivation

    1. pow2N/A

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

    2. lift-*.f6452.4

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

  5. Applied rewrites52.4%

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

Reproduce

?
herbie shell --seed 2025073 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))