FastMath test5

Average Error: 0.1 → 0

Time: 43.6s

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 r25827468 = d1;
        double r25827469 = r25827468 * r25827468;
        double r25827470 = r25827468 * r25827469;
        double r25827471 = r25827470 * r25827468;
        double r25827472 = r25827471 * r25827468;
        double r25827473 = r25827472 * r25827469;
        double r25827474 = r25827473 * r25827468;
        double r25827475 = r25827468 * r25827474;
        double r25827476 = r25827475 * r25827468;
        return r25827476;
}

↓

double f(double d1) {
        double r25827477 = d1;
        double r25827478 = 10.0;
        double r25827479 = pow(r25827477, r25827478);
        return r25827479;
}

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 pow1 0.1

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

5. Applied pow1 0.1

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

6. Applied pow-prod-up 0.1

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

7. Applied pow1 0.1

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

8. Applied pow1 0.1

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

9. Applied pow1 0.1

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

10. Applied pow-sqr 0.1

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

11. Applied pow1 0.1

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

12. Applied pow-prod-up 0.1

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

13. Applied pow-prod-up 0.1

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

14. Applied pow-plus 0.1

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

15. Applied pow-prod-up 0.1

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

16. Applied pow-prod-up 0.1

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

17. Simplified 0.1

    \[\leadsto \left(d1 \cdot {d1}^{\color{blue}{8}}\right) \cdot d1\]
18. Using strategy rm

19. Applied pow1 0.1

    \[\leadsto \left(d1 \cdot {d1}^{8}\right) \cdot \color{blue}{{d1}^{1}}\]

20. Applied pow1 0.1

    \[\leadsto \left(\color{blue}{{d1}^{1}} \cdot {d1}^{8}\right) \cdot {d1}^{1}\]

21. Applied pow-prod-up 0.0

    \[\leadsto \color{blue}{{d1}^{\left(1 + 8\right)}} \cdot {d1}^{1}\]

22. Applied pow-prod-up 0

    \[\leadsto \color{blue}{{d1}^{\left(\left(1 + 8\right) + 1\right)}}\]

23. Simplified 0

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

24. Final simplification 0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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