Average Error: 0.0 → 0.0
Time: 9.7s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\left(im + \left(-im\right)\right) \cdot im + \left(im + re\right) \cdot \left(re - im\right)\]
re \cdot re - im \cdot im
↓
\left(im + \left(-im\right)\right) \cdot im + \left(im + re\right) \cdot \left(re - im\right)
double f(double re, double im) {
        double r9517 = re;
        double r9518 = r9517 * r9517;
        double r9519 = im;
        double r9520 = r9519 * r9519;
        double r9521 = r9518 - r9520;
        return r9521;
}
↓
double f(double re, double im) {
        double r9522 = im;
        double r9523 = -r9522;
        double r9524 = r9522 + r9523;
        double r9525 = r9524 * r9522;
        double r9526 = re;
        double r9527 = r9522 + r9526;
        double r9528 = r9526 - r9522;
        double r9529 = r9527 * r9528;
        double r9530 = r9525 + r9529;
        return r9530;
}

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 prod-diff0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))