math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 36.5s

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 r353778 = re;
        double r353779 = r353778 * r353778;
        double r353780 = im;
        double r353781 = r353780 * r353780;
        double r353782 = r353779 - r353781;
        return r353782;
}

↓

double f(double re, double im) {
        double r353783 = re;
        double r353784 = im;
        double r353785 = r353783 - r353784;
        double r353786 = r353784 + r353783;
        double r353787 = r353785 * r353786;
        return r353787;
}

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 2019152 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))