math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 36.6s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r471226 = re;
        double r471227 = r471226 * r471226;
        double r471228 = im;
        double r471229 = r471228 * r471228;
        double r471230 = r471227 - r471229;
        return r471230;
}

↓

double f(double re, double im) {
        double r471231 = re;
        double r471232 = im;
        double r471233 = r471231 - r471232;
        double r471234 = r471232 + r471231;
        double r471235 = r471233 * r471234;
        return r471235;
}

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

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))