Average Error: 0 → 0
Time: 398.0ms
Precision: 64
\[2 \cdot \left(\left(1 \cdot \frac{1}{9} + \frac{1}{9} \cdot \frac{1}{9}\right) + \frac{1}{9} \cdot 1\right)\]
↓
\[\left(\mathsf{fma}\left(2, 1, \frac{1}{9}\right) \cdot \frac{2}{9}\right) \cdot 1\]
2 \cdot \left(\left(1 \cdot \frac{1}{9} + \frac{1}{9} \cdot \frac{1}{9}\right) + \frac{1}{9} \cdot 1\right)
↓
\left(\mathsf{fma}\left(2, 1, \frac{1}{9}\right) \cdot \frac{2}{9}\right) \cdot 1
double f() {
        double r104685 = 2.0;
        double r104686 = 1.0;
        double r104687 = 9.0;
        double r104688 = r104686 / r104687;
        double r104689 = r104686 * r104688;
        double r104690 = r104688 * r104688;
        double r104691 = r104689 + r104690;
        double r104692 = r104688 * r104686;
        double r104693 = r104691 + r104692;
        double r104694 = r104685 * r104693;
        return r104694;
}
↓
double f() {
        double r104695 = 2.0;
        double r104696 = 1.0;
        double r104697 = 9.0;
        double r104698 = r104696 / r104697;
        double r104699 = fma(r104695, r104696, r104698);
        double r104700 = 2.0;
        double r104701 = r104700 / r104697;
        double r104702 = r104699 * r104701;
        double r104703 = r104702 * r104696;
        return r104703;
}

Error

Target

Original0
Target0
Herbie0
\[\left(\left(\frac{1}{9} \cdot 1\right) \cdot 2 + 2 \cdot \left(\frac{1}{9} \cdot \frac{1}{9}\right)\right) + 2 \cdot \left(1 \cdot \frac{1}{9}\right)\]

Derivation

  1. Initial program 0

    \[2 \cdot \left(\left(1 \cdot \frac{1}{9} + \frac{1}{9} \cdot \frac{1}{9}\right) + \frac{1}{9} \cdot 1\right)\]

  2. Simplified0

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, 1, \frac{1}{9}\right) \cdot \frac{2}{9}\right) \cdot 1}\]

  3. Final simplification0

    \[\leadsto \left(\mathsf{fma}\left(2, 1, \frac{1}{9}\right) \cdot \frac{2}{9}\right) \cdot 1\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore ()
  :name "Rectangular parallelepiped of dimension a×b×c"
  :precision binary64

  :herbie-target
  (+ (+ (* (* (/ 1 9) 1) 2) (* 2 (* (/ 1 9) (/ 1 9)))) (* 2 (* 1 (/ 1 9))))

  (* 2 (+ (+ (* 1 (/ 1 9)) (* (/ 1 9) (/ 1 9))) (* (/ 1 9) 1))))