\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.001677300000000000058247850986958837893326 \land 0.0 \le d \le 0.001677300000000000058247850986958837893326\]

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

↓

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

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

↓

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

double f(double a, double b, double c, double d) {
        double r113477 = a;
        double r113478 = b;
        double r113479 = c;
        double r113480 = r113478 + r113479;
        double r113481 = d;
        double r113482 = r113480 + r113481;
        double r113483 = r113477 * r113482;
        return r113483;
}

↓

double f(double a, double b, double c, double d) {
        double r113484 = b;
        double r113485 = c;
        double r113486 = r113484 + r113485;
        double r113487 = a;
        double r113488 = r113486 * r113487;
        double r113489 = d;
        double r113490 = r113489 * r113487;
        double r113491 = r113488 + r113490;
        return r113491;
}

Error

The X axis uses an exponential scale — Bits error versus `a`

Original	0.0
Target	0.0
Herbie	0.0

Derivation

Initial program 0.0
\[a \cdot \left(\left(b + c\right) + d\right)\]
Using strategy rm
Applied distribute-lft-in0.0
\[\leadsto \color{blue}{a \cdot \left(b + c\right) + a \cdot d}\]
Simplified0.0
\[\leadsto \color{blue}{\left(b + c\right) \cdot a} + a \cdot d\]
Simplified0.0
\[\leadsto \left(b + c\right) \cdot a + \color{blue}{d \cdot a}\]
Final simplification0.0
\[\leadsto \left(b + c\right) \cdot a + d \cdot a\]

Reproduce

herbie shell --seed 2019350 +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)))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Try it out

Target

Derivation

Reproduce

In
Out