FastMath repmul

Average Error: 0.1 → 0

Time: 7.1s

Precision: 64

Math

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

↓

\[{d1}^{4}\]

C

double f(double d1) {
        double r8403965 = d1;
        double r8403966 = r8403965 * r8403965;
        double r8403967 = r8403966 * r8403965;
        double r8403968 = r8403967 * r8403965;
        return r8403968;
}

↓

double f(double d1) {
        double r8403969 = d1;
        double r8403970 = 4.0;
        double r8403971 = pow(r8403969, r8403970);
        return r8403971;
}

Error

Bits error versus d1

Target

Original	0.1
Target	0
Herbie	0

\[{d1}^{4}\]

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 2019146 +o rules:numerics
(FPCore (d1)
  :name "FastMath repmul"

  :herbie-target
  (pow d1 4)

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