math.square on complex, real part

Average Error: 0.0 → 0.0

Time: 1.0s

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 r570 = re;
        double r571 = r570 * r570;
        double r572 = im;
        double r573 = r572 * r572;
        double r574 = r571 - r573;
        return r574;
}

↓

double f(double re, double im) {
        double r575 = re;
        double r576 = im;
        double r577 = r576 * r576;
        double r578 = -r577;
        double r579 = fma(r575, r575, r578);
        return r579;
}

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 2020036 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))