FastMath test5

Average Error: 0.1 → 0

Time: 12.9s

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 r10326552 = d1;
        double r10326553 = r10326552 * r10326552;
        double r10326554 = r10326552 * r10326553;
        double r10326555 = r10326554 * r10326552;
        double r10326556 = r10326555 * r10326552;
        double r10326557 = r10326556 * r10326553;
        double r10326558 = r10326557 * r10326552;
        double r10326559 = r10326552 * r10326558;
        double r10326560 = r10326559 * r10326552;
        return r10326560;
}

↓

double f(double d1) {
        double r10326561 = d1;
        double r10326562 = 10.0;
        double r10326563 = pow(r10326561, r10326562);
        return r10326563;
}

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. Simplified 0.2

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

4. Applied pow1 0.2

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

5. Applied pow1 0.2

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

6. Applied pow-sqr 0.2

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

7. Applied pow1 0.2

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

8. Applied pow1 0.2

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

9. Applied pow-sqr 0.2

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

10. Applied pow1 0.2

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

11. Applied pow-prod-up 0.2

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

12. Applied pow-prod-up 0.1

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

13. Simplified 0.1

    \[\leadsto {d1}^{\color{blue}{5}} \cdot \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right)\]
14. Using strategy rm

15. Applied pow1 0.1

    \[\leadsto {d1}^{5} \cdot \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot \color{blue}{{d1}^{1}}\right)\right)\]

16. Applied pow1 0.1

    \[\leadsto {d1}^{5} \cdot \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\color{blue}{{d1}^{1}} \cdot {d1}^{1}\right)\right)\]

17. Applied pow-sqr 0.1

    \[\leadsto {d1}^{5} \cdot \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \color{blue}{{d1}^{\left(2 \cdot 1\right)}}\right)\]

18. Applied pow2 0.1

    \[\leadsto {d1}^{5} \cdot \left(\left(d1 \cdot \color{blue}{{d1}^{2}}\right) \cdot {d1}^{\left(2 \cdot 1\right)}\right)\]

19. Applied pow1 0.1

    \[\leadsto {d1}^{5} \cdot \left(\left(\color{blue}{{d1}^{1}} \cdot {d1}^{2}\right) \cdot {d1}^{\left(2 \cdot 1\right)}\right)\]

20. Applied pow-prod-up 0.1

    \[\leadsto {d1}^{5} \cdot \left(\color{blue}{{d1}^{\left(1 + 2\right)}} \cdot {d1}^{\left(2 \cdot 1\right)}\right)\]

21. Applied pow-prod-up 0.1

    \[\leadsto {d1}^{5} \cdot \color{blue}{{d1}^{\left(\left(1 + 2\right) + 2 \cdot 1\right)}}\]

22. Applied pow-prod-up 0

    \[\leadsto \color{blue}{{d1}^{\left(5 + \left(\left(1 + 2\right) + 2 \cdot 1\right)\right)}}\]

23. Simplified 0

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

24. Final simplification 0

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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