FastMath test3

Average Error: 0.1 → 0.1

Time: 9.0s

Precision: 64

Math

\[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]

↓

\[\left(d3 + \left(3 + d2\right)\right) \cdot d1\]

C

double f(double d1, double d2, double d3) {
        double r4705519 = d1;
        double r4705520 = 3.0;
        double r4705521 = r4705519 * r4705520;
        double r4705522 = d2;
        double r4705523 = r4705519 * r4705522;
        double r4705524 = r4705521 + r4705523;
        double r4705525 = d3;
        double r4705526 = r4705519 * r4705525;
        double r4705527 = r4705524 + r4705526;
        return r4705527;
}

↓

double f(double d1, double d2, double d3) {
        double r4705528 = d3;
        double r4705529 = 3.0;
        double r4705530 = d2;
        double r4705531 = r4705529 + r4705530;
        double r4705532 = r4705528 + r4705531;
        double r4705533 = d1;
        double r4705534 = r4705532 * r4705533;
        return r4705534;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Target

Original	0.1
Target	0.1
Herbie	0.1

\[d1 \cdot \left(\left(3 + d2\right) + d3\right)\]

Derivation

1. Initial program 0.1

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]

2. Simplified 0.1

   \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)}\]
3. Using strategy rm

4. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{d1 \cdot d2 + d1 \cdot \left(3 + d3\right)}\]

5. Taylor expanded around 0 0.1

   \[\leadsto \color{blue}{d2 \cdot d1 + \left(d3 \cdot d1 + 3 \cdot d1\right)}\]

6. Simplified 0.1

   \[\leadsto \color{blue}{d1 \cdot \left(d3 + \left(d2 + 3\right)\right)}\]

7. Final simplification 0.1

   \[\leadsto \left(d3 + \left(3 + d2\right)\right) \cdot d1\]

Reproduce

herbie shell --seed 2019153 
(FPCore (d1 d2 d3)
  :name "FastMath test3"

  :herbie-target
  (* d1 (+ (+ 3 d2) d3))

  (+ (+ (* d1 3) (* d1 d2)) (* d1 d3)))