Expanding a square

Average Error: 39.5 → 0.0

Time: 20.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

↓

\[x \cdot \left(2 + x\right)\]

C

double f(double x) {
        double r3393 = x;
        double r3394 = 1.0;
        double r3395 = r3393 + r3394;
        double r3396 = r3395 * r3395;
        double r3397 = r3396 - r3394;
        return r3397;
}

↓

double f(double x) {
        double r3398 = x;
        double r3399 = 2.0;
        double r3400 = r3399 + r3398;
        double r3401 = r3398 * r3400;
        return r3401;
}

Error

Bits error versus x

Derivation

1. Initial program 39.5

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

2. Taylor expanded around 0 0.0

   \[\leadsto \color{blue}{{x}^{2} + 2 \cdot x}\]

3. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right)}\]

4. Final simplification 0.0

   \[\leadsto x \cdot \left(2 + x\right)\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))