FastMath test5

Average Error: 0.1 → 0

Time: 15.3s

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 r9083035 = d1;
        double r9083036 = r9083035 * r9083035;
        double r9083037 = r9083035 * r9083036;
        double r9083038 = r9083037 * r9083035;
        double r9083039 = r9083038 * r9083035;
        double r9083040 = r9083039 * r9083036;
        double r9083041 = r9083040 * r9083035;
        double r9083042 = r9083035 * r9083041;
        double r9083043 = r9083042 * r9083035;
        return r9083043;
}

↓

double f(double d1) {
        double r9083044 = d1;
        double r9083045 = 10.0;
        double r9083046 = pow(r9083044, r9083045);
        return r9083046;
}

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

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

8. Applied pow1 0.1

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

9. Applied pow1 0.1

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

10. Applied pow1 0.1

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

11. Applied pow-prod-up 0.1

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

12. Applied pow1 0.1

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

13. Applied pow-prod-up 0.1

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

14. Applied pow-prod-up 0.1

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

15. Applied pow-prod-up 0.1

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

16. Applied pow-prod-up 0.1

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

17. Applied pow-prod-up 0.1

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

18. Simplified 0.1

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

20. Applied pow1 0.1

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

21. Applied pow1 0.1

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

22. Applied pow-prod-up 0.0

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

23. Applied pow-prod-up 0

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

24. Simplified 0

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

25. Final simplification 0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10.0)

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