math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 24.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r735499 = re;
        double r735500 = r735499 * r735499;
        double r735501 = im;
        double r735502 = r735501 * r735501;
        double r735503 = r735500 - r735502;
        return r735503;
}

↓

double f(double re, double im) {
        double r735504 = re;
        double r735505 = im;
        double r735506 = r735504 - r735505;
        double r735507 = r735505 + r735504;
        double r735508 = r735506 * r735507;
        return r735508;
}

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

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))