Expression, p14

Average Error: 0.0 → 0.0

Time: 8.9s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r7784250 = a;
        double r7784251 = b;
        double r7784252 = c;
        double r7784253 = r7784251 + r7784252;
        double r7784254 = d;
        double r7784255 = r7784253 + r7784254;
        double r7784256 = r7784250 * r7784255;
        return r7784256;
}

↓

double f(double a, double b, double c, double d) {
        double r7784257 = d;
        double r7784258 = a;
        double r7784259 = r7784257 * r7784258;
        double r7784260 = b;
        double r7784261 = c;
        double r7784262 = r7784261 * r7784258;
        double r7784263 = fma(r7784260, r7784258, r7784262);
        double r7784264 = r7784259 + r7784263;
        return r7784264;
}

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 distribute-rgt-in 0.0

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

7. Applied fma-def 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019119 +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)))