Average Error: 0.0 → 0.0
Time: 2.9s
Precision: 64
\[\left(x \cdot x + y\right) + y\]
↓
\[\mathsf{fma}\left(y, 2, x \cdot x\right)\]
\left(x \cdot x + y\right) + y
↓
\mathsf{fma}\left(y, 2, x \cdot x\right)
double f(double x, double y) {
        double r813309 = x;
        double r813310 = r813309 * r813309;
        double r813311 = y;
        double r813312 = r813310 + r813311;
        double r813313 = r813312 + r813311;
        return r813313;
}
↓
double f(double x, double y) {
        double r813314 = y;
        double r813315 = 2.0;
        double r813316 = x;
        double r813317 = r813316 * r813316;
        double r813318 = fma(r813314, r813315, r813317);
        return r813318;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2"
  :precision binary64

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

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