FastMath repmul

Average Error: 0.1 → 0

Time: 29.5s

Precision: 64

Math

\[\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\]

↓

\[{d1}^{4}\]

C

double f(double d1) {
        double r6175601 = d1;
        double r6175602 = r6175601 * r6175601;
        double r6175603 = r6175602 * r6175601;
        double r6175604 = r6175603 * r6175601;
        return r6175604;
}

↓

double f(double d1) {
        double r6175605 = d1;
        double r6175606 = 4.0;
        double r6175607 = pow(r6175605, r6175606);
        return r6175607;
}

Error

Bits error versus d1

Target

Original	0.1
Target	0
Herbie	0

\[{d1}^{4.0}\]

Derivation

1. Initial program 0.1

   \[\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\]
2. Using strategy rm

3. Applied pow1 0.1

   \[\leadsto \left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot \color{blue}{{d1}^{1}}\]

4. Applied pow1 0.1

   \[\leadsto \left(\left(d1 \cdot d1\right) \cdot \color{blue}{{d1}^{1}}\right) \cdot {d1}^{1}\]

5. Applied pow1 0.1

   \[\leadsto \left(\left(d1 \cdot \color{blue}{{d1}^{1}}\right) \cdot {d1}^{1}\right) \cdot {d1}^{1}\]

6. Applied pow1 0.1

   \[\leadsto \left(\left(\color{blue}{{d1}^{1}} \cdot {d1}^{1}\right) \cdot {d1}^{1}\right) \cdot {d1}^{1}\]

7. Applied pow-prod-up 0.1

   \[\leadsto \left(\color{blue}{{d1}^{\left(1 + 1\right)}} \cdot {d1}^{1}\right) \cdot {d1}^{1}\]

8. Applied pow-prod-up 0.1

   \[\leadsto \color{blue}{{d1}^{\left(\left(1 + 1\right) + 1\right)}} \cdot {d1}^{1}\]

9. Applied pow-prod-up 0

   \[\leadsto \color{blue}{{d1}^{\left(\left(\left(1 + 1\right) + 1\right) + 1\right)}}\]

10. Simplified 0

    \[\leadsto {d1}^{\color{blue}{4}}\]

11. Final simplification 0

    \[\leadsto {d1}^{4}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (d1)
  :name "FastMath repmul"

  :herbie-target
  (pow d1 4.0)

  (* (* (* d1 d1) d1) d1))