math.square on complex, real part

Average Error: 0.0 → 0.0

Time: 11.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double re, double im) {
        double r676 = re;
        double r677 = r676 * r676;
        double r678 = im;
        double r679 = r678 * r678;
        double r680 = r677 - r679;
        return r680;
}

↓

double f(double re, double im) {
        double r681 = re;
        double r682 = im;
        double r683 = r682 * r682;
        double r684 = -r683;
        double r685 = fma(r681, r681, r684);
        return r685;
}

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 prod-diff 0.0

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

4. Simplified 0.0

   \[\leadsto \mathsf{fma}\left(re, re, -im \cdot im\right) + \color{blue}{0}\]

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))