Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(a, d + b, c \cdot a\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(a, d + b, c \cdot a\right)
double f(double a, double b, double c, double d) {
        double r93362 = a;
        double r93363 = b;
        double r93364 = c;
        double r93365 = r93363 + r93364;
        double r93366 = d;
        double r93367 = r93365 + r93366;
        double r93368 = r93362 * r93367;
        return r93368;
}
↓
double f(double a, double b, double c, double d) {
        double r93369 = a;
        double r93370 = d;
        double r93371 = b;
        double r93372 = r93370 + r93371;
        double r93373 = c;
        double r93374 = r93373 * r93369;
        double r93375 = fma(r93369, r93372, r93374);
        return r93375;
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*r*0.6

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

  6. Simplified0.6

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

  7. Taylor expanded around inf 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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)))