Average Error: 0.0 → 0.0
Time: 1.4s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\mathsf{fma}\left(\sqrt{re \cdot re}, \sqrt{re \cdot re}, -im \cdot im\right)\]
re \cdot re - im \cdot im
↓
\mathsf{fma}\left(\sqrt{re \cdot re}, \sqrt{re \cdot re}, -im \cdot im\right)
double code(double re, double im) {
	return ((re * re) - (im * im));
}
↓
double code(double re, double im) {
	return fma(sqrt((re * re)), sqrt((re * re)), -(im * im));
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied fma-neg0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))