math.square on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%
Time: 1.2s
Alternatives: 1
Speedup: N/A×

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);
}

real(8) function im_sqr(re, im)
    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}

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 1 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);
}

real(8) function im_sqr(re, im)
    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, 1.6× speedup?

\[\begin{array}{l} [re, im] = \mathsf{sort}([re, im])\\ \\ im \cdot \left(re + re\right) \end{array} \]
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore im_sqr (re im) :precision binary64 (* im (+ re re)))
assert(re < im);
double im_sqr(double re, double im) {
	return im * (re + re);
}

NOTE: re and im should be sorted in increasing order before calling this function.
real(8) function im_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    im_sqr = im * (re + re)
end function

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

[re, im] = sort([re, im])
def im_sqr(re, im):
	return im * (re + re)

re, im = sort([re, im])
function im_sqr(re, im)
	return Float64(im * Float64(re + re))
end

re, im = num2cell(sort([re, im])){:}
function tmp = im_sqr(re, im)
	tmp = im * (re + re);
end

NOTE: re and im should be sorted in increasing order before calling this function.
im$95$sqr[re_, im_] := N[(im * N[(re + re), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
[re, im] = \mathsf{sort}([re, im])\\
\\
im \cdot \left(re + re\right)
\end{array}
Derivation

  1. Initial program 100.0%

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

    1. lift-+.f64N/A

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

    2. lift-*.f64N/A

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

    3. lift-*.f64N/A

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

    4. *-commutativeN/A

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

    5. distribute-rgt-outN/A

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

    6. lower-*.f64N/A

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

    7. lower-+.f6499.6

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

  4. Applied rewrites99.6%

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

Reproduce

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herbie shell --seed 2024308 
(FPCore im_sqr (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))