Average Error: 0.0 → 0.0
Time: 11.6s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(b + c, a, d \cdot a\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(b + c, a, d \cdot a\right)
double f(double a, double b, double c, double d) {
        double r65549 = a;
        double r65550 = b;
        double r65551 = c;
        double r65552 = r65550 + r65551;
        double r65553 = d;
        double r65554 = r65552 + r65553;
        double r65555 = r65549 * r65554;
        return r65555;
}
↓
double f(double a, double b, double c, double d) {
        double r65556 = b;
        double r65557 = c;
        double r65558 = r65556 + r65557;
        double r65559 = a;
        double r65560 = d;
        double r65561 = r65560 * r65559;
        double r65562 = fma(r65558, r65559, r65561);
        return r65562;
}

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 distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  7. Applied fma-def0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (<= 56789 a 98765) (<= 0.0 b 1) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

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