Average Error: 0.0 → 0.0
Time: 7.3s
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 r8633 = re;
        double r8634 = r8633 * r8633;
        double r8635 = im;
        double r8636 = r8635 * r8635;
        double r8637 = r8634 - r8636;
        return r8637;
}
↓
double f(double re, double im) {
        double r8638 = re;
        double r8639 = im;
        double r8640 = r8639 * r8639;
        double r8641 = -r8640;
        double r8642 = fma(r8638, r8638, r8641);
        return r8642;
}

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 2019198 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))