math.square on complex, real part

Average Error: 0.0 → 0.0

Time: 1.2s

Precision: 64

Math

\[re \cdot re - im \cdot im\]

↓

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

C

double f(double re, double im) {
        double r581 = re;
        double r582 = r581 * r581;
        double r583 = im;
        double r584 = r583 * r583;
        double r585 = r582 - r584;
        return r585;
}

↓

double f(double re, double im) {
        double r586 = re;
        double r587 = im;
        double r588 = r587 * r587;
        double r589 = -r588;
        double r590 = fma(r586, r586, r589);
        return r590;
}

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-neg 0.0

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

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))