math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 16.9s

Precision: 64

Math

\[\left(re \cdot re\right) - \left(im \cdot im\right)\]

↓

\[\left(re - im\right) \cdot \left(re + im\right)\]

C

double f(double re, double im) {
        double r593583 = re;
        double r593584 = r593583 * r593583;
        double r593585 = im;
        double r593586 = r593585 * r593585;
        double r593587 = r593584 - r593586;
        return r593587;
}

↓

double f(double re, double im) {
        double r593588 = re;
        double r593589 = im;
        double r593590 = r593588 - r593589;
        double r593591 = r593588 + r593589;
        double r593592 = r593590 * r593591;
        return r593592;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.3

   \[\left(re \cdot re\right) - \left(im \cdot im\right)\]

2. Simplified 0.1

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

3. Final simplification 0.1

   \[\leadsto \left(re - im\right) \cdot \left(re + im\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))