Average Error: 0.0 → 0.0
Time: 14.0s
Precision: 64
\[a \cdot a - b \cdot b\]
↓
\[\mathsf{fma}\left(a, a, -b \cdot b\right)\]
a \cdot a - b \cdot b
↓
\mathsf{fma}\left(a, a, -b \cdot b\right)
double f(double a, double b) {
        double r61608 = a;
        double r61609 = r61608 * r61608;
        double r61610 = b;
        double r61611 = r61610 * r61610;
        double r61612 = r61609 - r61611;
        return r61612;
}
↓
double f(double a, double b) {
        double r61613 = a;
        double r61614 = b;
        double r61615 = r61614 * r61614;
        double r61616 = -r61615;
        double r61617 = fma(r61613, r61613, r61616);
        return r61617;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0.0
\[\left(a + b\right) \cdot \left(a - b\right)\]

Derivation

  1. Initial program 0.0

    \[a \cdot a - b \cdot b\]
  2. Using strategy rm

  3. Applied prod-diff0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right) + \mathsf{fma}\left(-b, b, b \cdot b\right)}\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(a, a, -b \cdot b\right) + \color{blue}{0}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(a, a, -b \cdot b\right)\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))