Average Error: 0.0 → 0.0
Time: 865.0ms
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 r877275 = x;
        double r877276 = r877275 * r877275;
        double r877277 = y;
        double r877278 = r877276 + r877277;
        double r877279 = r877278 + r877277;
        return r877279;
}
↓
double f(double x, double y) {
        double r877280 = y;
        double r877281 = 2.0;
        double r877282 = x;
        double r877283 = pow(r877282, r877281);
        double r877284 = fma(r877280, r877281, r877283);
        return r877284;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto y + \color{blue}{1 \cdot \mathsf{fma}\left(x, x, y\right)}\]

  5. Applied *-un-lft-identity0.0

    \[\leadsto \color{blue}{1 \cdot y} + 1 \cdot \mathsf{fma}\left(x, x, y\right)\]

  6. Applied distribute-lft-out0.0

    \[\leadsto \color{blue}{1 \cdot \left(y + \mathsf{fma}\left(x, x, y\right)\right)}\]

  7. Simplified0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(y, 2, {x}^{2}\right)}\]

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +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))