Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\mathsf{fma}\left(re, re, -im \cdot im\right)\]
re \cdot re - im \cdot im
↓
\mathsf{fma}\left(re, re, -im \cdot im\right)
double f(double re, double im) {
        double r1734 = re;
        double r1735 = r1734 * r1734;
        double r1736 = im;
        double r1737 = r1736 * r1736;
        double r1738 = r1735 - r1737;
        return r1738;
}
↓
double f(double re, double im) {
        double r1739 = re;
        double r1740 = im;
        double r1741 = r1740 * r1740;
        double r1742 = -r1741;
        double r1743 = fma(r1739, r1739, r1742);
        return r1743;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))