math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 31.5s

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 r407067 = re;
        double r407068 = r407067 * r407067;
        double r407069 = im;
        double r407070 = r407069 * r407069;
        double r407071 = r407068 - r407070;
        return r407071;
}

↓

double f(double re, double im) {
        double r407072 = re;
        double r407073 = im;
        double r407074 = r407072 - r407073;
        double r407075 = r407073 + r407072;
        double r407076 = r407074 * r407075;
        return r407076;
}

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 2019142 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))