Average Error: 0.0 → 0.0
Time: 1.0s
Precision: binary64
\[re \cdot re + im \cdot im \]
↓
\[\mathsf{hypot}\left(im, re\right) \cdot \mathsf{hypot}\left(im, re\right) \]
(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))
↓
(FPCore modulus_sqr (re im)
 :precision binary64
 (* (hypot im re) (hypot im re)))
double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}
↓
double modulus_sqr(double re, double im) {
	return hypot(im, re) * hypot(im, re);
}

public static double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}
↓
public static double modulus_sqr(double re, double im) {
	return Math.hypot(im, re) * Math.hypot(im, re);
}

def modulus_sqr(re, im):
	return (re * re) + (im * im)
↓
def modulus_sqr(re, im):
	return math.hypot(im, re) * math.hypot(im, re)

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

function tmp = modulus_sqr(re, im)
	tmp = (re * re) + (im * im);
end
↓
function tmp = modulus_sqr(re, im)
	tmp = hypot(im, re) * hypot(im, re);
end

modulus$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]
↓
modulus$95$sqr[re_, im_] := N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] * N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Applied add-sqr-sqrt_binary640.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022137 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))