Average Error: 0.0 → 0.0
Time: 10.0s
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 r9504 = re;
        double r9505 = r9504 * r9504;
        double r9506 = im;
        double r9507 = r9506 * r9506;
        double r9508 = r9505 - r9507;
        return r9508;
}
↓
double f(double re, double im) {
        double r9509 = im;
        double r9510 = -r9509;
        double r9511 = r9509 + r9510;
        double r9512 = r9511 * r9509;
        double r9513 = re;
        double r9514 = r9509 + r9513;
        double r9515 = r9513 - r9509;
        double r9516 = r9514 * r9515;
        double r9517 = r9512 + r9516;
        return r9517;
}

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 2019179 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))