Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right)\]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right)
double f(double x, double y) {
        double r354120 = x;
        double r354121 = 2.0;
        double r354122 = r354120 * r354121;
        double r354123 = r354120 * r354120;
        double r354124 = r354122 + r354123;
        double r354125 = y;
        double r354126 = r354125 * r354125;
        double r354127 = r354124 + r354126;
        return r354127;
}
↓
double f(double x, double y) {
        double r354128 = y;
        double r354129 = x;
        double r354130 = 2.0;
        double r354131 = r354130 + r354129;
        double r354132 = r354129 * r354131;
        double r354133 = fma(r354128, r354128, r354132);
        return r354133;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[y \cdot y + \left(2 \cdot x + x \cdot x\right)\]

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2 x) (* x x)))

  (+ (+ (* x 2) (* x x)) (* y y)))