math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 11.9s

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 r626137 = re;
        double r626138 = r626137 * r626137;
        double r626139 = im;
        double r626140 = r626139 * r626139;
        double r626141 = r626138 - r626140;
        return r626141;
}

↓

double f(double re, double im) {
        double r626142 = re;
        double r626143 = im;
        double r626144 = r626142 - r626143;
        double r626145 = r626143 + r626142;
        double r626146 = r626144 * r626145;
        return r626146;
}

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 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))