Average Error: 0.0 → 0.0
Time: 18.9s
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 r8652 = re;
        double r8653 = r8652 * r8652;
        double r8654 = im;
        double r8655 = r8654 * r8654;
        double r8656 = r8653 - r8655;
        return r8656;
}
↓
double f(double re, double im) {
        double r8657 = re;
        double r8658 = im;
        double r8659 = r8658 * r8658;
        double r8660 = -r8659;
        double r8661 = fma(r8657, r8657, r8660);
        return r8661;
}

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 2019326 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))