FastMath test5

Average Error: 0.1 → 0

Time: 28.9s

Precision: 64

Math

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

↓

\[{d1}^{10}\]

C

double f(double d1) {
        double r25157612 = d1;
        double r25157613 = r25157612 * r25157612;
        double r25157614 = r25157612 * r25157613;
        double r25157615 = r25157614 * r25157612;
        double r25157616 = r25157615 * r25157612;
        double r25157617 = r25157616 * r25157613;
        double r25157618 = r25157617 * r25157612;
        double r25157619 = r25157612 * r25157618;
        double r25157620 = r25157619 * r25157612;
        return r25157620;
}

↓

double f(double d1) {
        double r25157621 = d1;
        double r25157622 = 10.0;
        double r25157623 = pow(r25157621, r25157622);
        return r25157623;
}

Error

Bits error versus d1

Target

Original	0.1
Target	0
Herbie	0

\[{d1}^{10}\]

Derivation

1. Initial program 0.1

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

3. Applied pow1 0.1

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

4. Applied pow2 0.1

   \[\leadsto \left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \color{blue}{{d1}^{2}}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

5. Applied pow1 0.1

   \[\leadsto \left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \color{blue}{{d1}^{1}}\right) \cdot {d1}^{2}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

6. Applied pow2 0.1

   \[\leadsto \left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \color{blue}{{d1}^{2}}\right) \cdot d1\right) \cdot {d1}^{1}\right) \cdot {d1}^{2}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

7. Applied pow1 0.1

   \[\leadsto \left(d1 \cdot \left(\left(\left(\left(\left(\color{blue}{{d1}^{1}} \cdot {d1}^{2}\right) \cdot d1\right) \cdot {d1}^{1}\right) \cdot {d1}^{2}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

8. Applied pow-prod-up 0.1

   \[\leadsto \left(d1 \cdot \left(\left(\left(\left(\color{blue}{{d1}^{\left(1 + 2\right)}} \cdot d1\right) \cdot {d1}^{1}\right) \cdot {d1}^{2}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

9. Applied pow-plus 0.1

   \[\leadsto \left(d1 \cdot \left(\left(\left(\color{blue}{{d1}^{\left(\left(1 + 2\right) + 1\right)}} \cdot {d1}^{1}\right) \cdot {d1}^{2}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

10. Applied pow-prod-up 0.1

    \[\leadsto \left(d1 \cdot \left(\left(\color{blue}{{d1}^{\left(\left(\left(1 + 2\right) + 1\right) + 1\right)}} \cdot {d1}^{2}\right) \cdot {d1}^{1}\right)\right) \cdot d1\]

11. Applied pow-prod-up 0.1

    \[\leadsto \left(d1 \cdot \left(\color{blue}{{d1}^{\left(\left(\left(\left(1 + 2\right) + 1\right) + 1\right) + 2\right)}} \cdot {d1}^{1}\right)\right) \cdot d1\]

12. Applied pow-prod-up 0.1

    \[\leadsto \left(d1 \cdot \color{blue}{{d1}^{\left(\left(\left(\left(\left(1 + 2\right) + 1\right) + 1\right) + 2\right) + 1\right)}}\right) \cdot d1\]

13. Simplified 0.1

    \[\leadsto \left(d1 \cdot {d1}^{\color{blue}{8}}\right) \cdot d1\]

14. Taylor expanded around inf 0

    \[\leadsto \color{blue}{{d1}^{10}}\]

15. Final simplification 0

    \[\leadsto {d1}^{10}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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