Average Error: 0.0 → 0.0
Time: 8.4s
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 r18227042 = x;
        double r18227043 = 2.0;
        double r18227044 = r18227042 * r18227043;
        double r18227045 = r18227042 * r18227042;
        double r18227046 = r18227044 + r18227045;
        double r18227047 = y;
        double r18227048 = r18227047 * r18227047;
        double r18227049 = r18227046 + r18227048;
        return r18227049;
}
↓
double f(double x, double y) {
        double r18227050 = y;
        double r18227051 = x;
        double r18227052 = 2.0;
        double r18227053 = r18227052 + r18227051;
        double r18227054 = r18227051 * r18227053;
        double r18227055 = fma(r18227050, r18227050, r18227054);
        return r18227055;
}

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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))