FastMath test5

Average Error: 0.1 → 0

Time: 14.0s

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 r3834294 = d1;
        double r3834295 = r3834294 * r3834294;
        double r3834296 = r3834294 * r3834295;
        double r3834297 = r3834296 * r3834294;
        double r3834298 = r3834297 * r3834294;
        double r3834299 = r3834298 * r3834295;
        double r3834300 = r3834299 * r3834294;
        double r3834301 = r3834294 * r3834300;
        double r3834302 = r3834301 * r3834294;
        return r3834302;
}

↓

double f(double d1) {
        double r3834303 = d1;
        double r3834304 = 10.0;
        double r3834305 = pow(r3834303, r3834304);
        return r3834305;
}

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-sqr 0.1

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

7. Applied pow1 0.1

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

8. Applied pow2 0.1

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

9. Applied pow1 0.1

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

10. Applied pow-prod-up 0.1

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

11. Applied pow-plus 0.1

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

12. Applied pow-prod-up 0.1

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

13. Applied pow-prod-up 0.1

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

14. Applied pow-prod-up 0.1

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

15. Simplified 0.1

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

17. Applied pow1 0.1

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

18. Applied pow1 0.1

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

19. Applied pow-prod-up 0.1

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

20. Applied pow-prod-up 0

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

21. Simplified 0

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

22. Final simplification 0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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