Average Error: 0.0 → 0.0
Time: 4.9s
Precision: 64
\[\left(x \cdot x + y\right) + y\]
↓
\[\mathsf{fma}\left(y, 2, {x}^{2}\right)\]
\left(x \cdot x + y\right) + y
↓
\mathsf{fma}\left(y, 2, {x}^{2}\right)
double f(double x, double y) {
        double r632452 = x;
        double r632453 = r632452 * r632452;
        double r632454 = y;
        double r632455 = r632453 + r632454;
        double r632456 = r632455 + r632454;
        return r632456;
}
↓
double f(double x, double y) {
        double r632457 = y;
        double r632458 = 2.0;
        double r632459 = x;
        double r632460 = pow(r632459, r632458);
        double r632461 = fma(r632457, r632458, r632460);
        return r632461;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\left(y + y\right) + x \cdot x\]

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y\right) + y}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, {x}^{2}\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, 2, {x}^{2}\right)\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2"

  :herbie-target
  (+ (+ y y) (* x x))

  (+ (+ (* x x) y) y))