Expression, p14

Average Error: 0.0 → 0.0

Time: 8.0s

Precision: 64

\[56789 \le a \le 98765 \land 0 \le b \le 1 \land 0 \le c \le 0.0016773 \land 0 \le d \le 0.0016773\]

Math

\[a \cdot \left(\left(b + c\right) + d\right)\]

↓

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

C

double f(double a, double b, double c, double d) {
        double r16032411 = a;
        double r16032412 = b;
        double r16032413 = c;
        double r16032414 = r16032412 + r16032413;
        double r16032415 = d;
        double r16032416 = r16032414 + r16032415;
        double r16032417 = r16032411 * r16032416;
        return r16032417;
}

↓

double f(double a, double b, double c, double d) {
        double r16032418 = a;
        double r16032419 = b;
        double r16032420 = c;
        double r16032421 = r16032419 + r16032420;
        double r16032422 = d;
        double r16032423 = r16032422 * r16032418;
        double r16032424 = fma(r16032418, r16032421, r16032423);
        return r16032424;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	0.0
Target	0.0
Herbie	0.0

\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

1. Initial program 0.0

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

3. Applied distribute-lft-in 0.0

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

5. Applied fma-def 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019120 +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)))