Average Error: 0.0 → 0.0
Time: 11.4s
Precision: 64
\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\mathsf{fma}\left(a, d, \left(c + b\right) \cdot a\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
\mathsf{fma}\left(a, d, \left(c + b\right) \cdot a\right)
double f(double a, double b, double c, double d) {
        double r1515615 = a;
        double r1515616 = b;
        double r1515617 = c;
        double r1515618 = r1515616 + r1515617;
        double r1515619 = d;
        double r1515620 = r1515618 + r1515619;
        double r1515621 = r1515615 * r1515620;
        return r1515621;
}
↓
double f(double a, double b, double c, double d) {
        double r1515622 = a;
        double r1515623 = d;
        double r1515624 = c;
        double r1515625 = b;
        double r1515626 = r1515624 + r1515625;
        double r1515627 = r1515626 * r1515622;
        double r1515628 = fma(r1515622, r1515623, r1515627);
        return r1515628;
}

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. Using strategy rm

  5. Applied flip-+5.1

    \[\leadsto a \cdot \color{blue}{\frac{b \cdot b - c \cdot c}{b - c}} + a \cdot d\]

  6. Applied associate-*r/5.2

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

  7. Taylor expanded around -inf 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789 a 98765) (<= 0 b 1) (<= 0 c 0.0016773) (<= 0 d 0.0016773))

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

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