math.square on complex, real part

Average Accuracy: 100.0% → 100.0%

Time: 1.4s

Precision: binary64

Cost: 448

Math

\[re \cdot re - im \cdot im \]

↓

\[\left(re - im\right) \cdot \left(re + im\right) \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

↓

(FPCore re_sqr (re im) :precision binary64 (* (- re im) (+ re im)))

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

   \[re \cdot re - im \cdot im \]

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ re \cdot re - im \cdot im \]
   difference-of-squares [=>] 100.0	\[ \color{blue}{\left(re + im\right) \cdot \left(re - im\right)} \]
   *-commutative [=>] 100.0	\[ \color{blue}{\left(re - im\right) \cdot \left(re + im\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	79.2%
Cost	521

\[\begin{array}{l} \mathbf{if}\;im \leq -4.4 \cdot 10^{-125} \lor \neg \left(im \leq 1.05 \cdot 10^{-36}\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2
Accuracy	56.0%
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2023137 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))