Expression, p14

Average Error: 0.0 → 0.0

Time: 7.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.0016773000000000001 \land 0.0 \le d \le 0.0016773000000000001\]

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r73243 = a;
        double r73244 = b;
        double r73245 = c;
        double r73246 = r73244 + r73245;
        double r73247 = d;
        double r73248 = r73246 + r73247;
        double r73249 = r73243 * r73248;
        return r73249;
}

↓

double f(double a, double b, double c, double d) {
        double r73250 = b;
        double r73251 = c;
        double r73252 = r73250 + r73251;
        double r73253 = a;
        double r73254 = d;
        double r73255 = r73253 * r73254;
        double r73256 = fma(r73252, r73253, r73255);
        return r73256;
}

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. Simplified 0.0

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

6. Applied fma-def 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020046 +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)))