Average Error: 0.0 → 0.0
Time: 18.1s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(c, a, a \cdot \left(b + d\right)\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(c, a, a \cdot \left(b + d\right)\right)
double f(double a, double b, double c, double d) {
        double r4539553 = a;
        double r4539554 = b;
        double r4539555 = c;
        double r4539556 = r4539554 + r4539555;
        double r4539557 = d;
        double r4539558 = r4539556 + r4539557;
        double r4539559 = r4539553 * r4539558;
        return r4539559;
}
↓
double f(double a, double b, double c, double d) {
        double r4539560 = c;
        double r4539561 = a;
        double r4539562 = b;
        double r4539563 = d;
        double r4539564 = r4539562 + r4539563;
        double r4539565 = r4539561 * r4539564;
        double r4539566 = fma(r4539560, r4539561, r4539565);
        return r4539566;
}

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

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

  5. Applied sqrt-prod0.5

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

  6. Taylor expanded around inf 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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)))