FastMath test5

Average Error: 0.1 → 0

Time: 53.2s

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 r6857954 = d1;
        double r6857955 = r6857954 * r6857954;
        double r6857956 = r6857954 * r6857955;
        double r6857957 = r6857956 * r6857954;
        double r6857958 = r6857957 * r6857954;
        double r6857959 = r6857958 * r6857955;
        double r6857960 = r6857959 * r6857954;
        double r6857961 = r6857954 * r6857960;
        double r6857962 = r6857961 * r6857954;
        return r6857962;
}

↓

double f(double d1) {
        double r6857963 = d1;
        double r6857964 = 10.0;
        double r6857965 = pow(r6857963, r6857964);
        return r6857965;
}

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

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

4. Applied pow1 0.1

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

5. Applied pow1 0.1

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

6. Applied pow1 0.1

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

7. Applied pow-prod-up 0.1

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

8. Applied pow1 0.1

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

9. Applied pow1 0.1

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

10. Applied pow1 0.1

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

11. Applied pow1 0.1

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

12. Applied pow-prod-up 0.1

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

13. Applied pow1 0.1

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

14. Applied pow-prod-up 0.1

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

15. Applied pow-prod-up 0.1

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

16. Applied pow-prod-up 0.1

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

17. Applied pow-prod-up 0.1

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

18. Applied pow-prod-up 0.1

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

19. Applied pow1 0.1

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

20. Applied pow-prod-up 0.0

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

21. Applied pow-prod-up 0

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

22. Simplified 0

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

23. Final simplification 0

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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