math.abs on complex (squared)

?

Percentage Accurate: 100.0% → 100.0%
Time: 3.2s
Precision: binary64
Cost: 6720

?

\[re \cdot re + im \cdot im \]
↓
\[\mathsf{fma}\left(re, re, im \cdot im\right) \]
(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))
↓
(FPCore modulus_sqr (re im) :precision binary64 (fma re re (* im im)))
double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}
↓
double modulus_sqr(double re, double im) {
	return fma(re, re, (im * im));
}

function modulus_sqr(re, im)
	return Float64(Float64(re * re) + Float64(im * im))
end
↓
function modulus_sqr(re, im)
	return fma(re, re, Float64(im * im))
end

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

re \cdot re + im \cdot im
↓
\mathsf{fma}\left(re, re, im \cdot im\right)

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.

Herbie found 4 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

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.

Bogosity?

Bogosity

Derivation?

  1. Initial program 100.0%

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

  2. Simplified100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)} \]
    Step-by-step derivation

    [Start]100.0%

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

    fma-def [=>]100.0%

    \[ \color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)} \]

  3. Final simplification100.0%

    \[\leadsto \mathsf{fma}\left(re, re, im \cdot im\right) \]

Alternatives

Alternative 1
Accuracy100.0%
Cost6720
\[\mathsf{fma}\left(re, re, im \cdot im\right) \]
Alternative 2
Accuracy100.0%
Cost448
\[im \cdot im + re \cdot re \]
Alternative 3
Accuracy68.8%
Cost324
\[\begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]
Alternative 4
Accuracy57.5%
Cost192
\[im \cdot im \]

Reproduce?

herbie shell --seed 2023263 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))