Average Error: 0.0 → 0.0
Time: 12.7s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\mathsf{fma}\left(re, re, \left(-im \cdot im\right)\right)\]
re \cdot re - im \cdot im
↓
\mathsf{fma}\left(re, re, \left(-im \cdot im\right)\right)
double f(double re, double im) {
        double r237606 = re;
        double r237607 = r237606 * r237606;
        double r237608 = im;
        double r237609 = r237608 * r237608;
        double r237610 = r237607 - r237609;
        return r237610;
}
↓
double f(double re, double im) {
        double r237611 = re;
        double r237612 = im;
        double r237613 = r237612 * r237612;
        double r237614 = -r237613;
        double r237615 = fma(r237611, r237611, r237614);
        return r237615;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[re \cdot re - im \cdot im\]
  2. Using strategy rm

  3. Applied fma-neg0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))