Average Error: 0.0 → 0.0
Time: 1.7s
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 r696435 = x;
        double r696436 = r696435 * r696435;
        double r696437 = y;
        double r696438 = r696436 + r696437;
        double r696439 = r696438 + r696437;
        return r696439;
}
↓
double f(double x, double y) {
        double r696440 = y;
        double r696441 = 2.0;
        double r696442 = x;
        double r696443 = pow(r696442, r696441);
        double r696444 = fma(r696440, r696441, r696443);
        return r696444;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))