math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 23.6s

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 r735562 = re;
        double r735563 = r735562 * r735562;
        double r735564 = im;
        double r735565 = r735564 * r735564;
        double r735566 = r735563 - r735565;
        return r735566;
}

↓

double f(double re, double im) {
        double r735567 = re;
        double r735568 = im;
        double r735569 = r735567 - r735568;
        double r735570 = r735568 + r735567;
        double r735571 = r735569 * r735570;
        return r735571;
}

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 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))