Average Error: 0.0 → 0.0
Time: 1.2s
Precision: 64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, {y}^{2}\right)\right)\]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, {y}^{2}\right)\right)
double f(double x, double y) {
        double r413377 = x;
        double r413378 = 2.0;
        double r413379 = r413377 * r413378;
        double r413380 = r413377 * r413377;
        double r413381 = r413379 + r413380;
        double r413382 = y;
        double r413383 = r413382 * r413382;
        double r413384 = r413381 + r413383;
        return r413384;
}
↓
double f(double x, double y) {
        double r413385 = x;
        double r413386 = 2.0;
        double r413387 = y;
        double r413388 = 2.0;
        double r413389 = pow(r413387, r413388);
        double r413390 = fma(r413386, r413385, r413389);
        double r413391 = fma(r413385, r413385, r413390);
        return r413391;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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)))