math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 16.0s

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 r1051139 = re;
        double r1051140 = r1051139 * r1051139;
        double r1051141 = im;
        double r1051142 = r1051141 * r1051141;
        double r1051143 = r1051140 - r1051142;
        return r1051143;
}

↓

double f(double re, double im) {
        double r1051144 = re;
        double r1051145 = im;
        double r1051146 = r1051144 - r1051145;
        double r1051147 = r1051145 + r1051144;
        double r1051148 = r1051146 * r1051147;
        return r1051148;
}

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 2019130 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))