Average Error: 0.0 → 0.0
Time: 15.0s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(c, a, b \cdot a + a \cdot d\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(c, a, b \cdot a + a \cdot d\right)
double f(double a, double b, double c, double d) {
        double r4471648 = a;
        double r4471649 = b;
        double r4471650 = c;
        double r4471651 = r4471649 + r4471650;
        double r4471652 = d;
        double r4471653 = r4471651 + r4471652;
        double r4471654 = r4471648 * r4471653;
        return r4471654;
}
↓
double f(double a, double b, double c, double d) {
        double r4471655 = c;
        double r4471656 = a;
        double r4471657 = b;
        double r4471658 = r4471657 * r4471656;
        double r4471659 = d;
        double r4471660 = r4471656 * r4471659;
        double r4471661 = r4471658 + r4471660;
        double r4471662 = fma(r4471655, r4471656, r4471661);
        return r4471662;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

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

Derivation

  1. Initial program 0.0

    \[a \cdot \left(\left(b + c\right) + d\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.5

    \[\leadsto a \cdot \color{blue}{\left(\sqrt{\left(b + c\right) + d} \cdot \sqrt{\left(b + c\right) + d}\right)}\]

  4. Applied associate-*r*0.5

    \[\leadsto \color{blue}{\left(a \cdot \sqrt{\left(b + c\right) + d}\right) \cdot \sqrt{\left(b + c\right) + d}}\]

  5. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{a \cdot d + \left(a \cdot b + a \cdot c\right)}\]

  6. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(c, a, a \cdot \left(b + d\right)\right)}\]
  7. Using strategy rm

  8. Applied distribute-lft-in0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789.0 a 98765.0) (<= 0.0 b 1.0) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

  (* a (+ (+ b c) d)))