FastMath test5

Average Error: 0.1 → 0

Time: 1.1m

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 r61317422 = d1;
        double r61317423 = r61317422 * r61317422;
        double r61317424 = r61317422 * r61317423;
        double r61317425 = r61317424 * r61317422;
        double r61317426 = r61317425 * r61317422;
        double r61317427 = r61317426 * r61317423;
        double r61317428 = r61317427 * r61317422;
        double r61317429 = r61317422 * r61317428;
        double r61317430 = r61317429 * r61317422;
        return r61317430;
}

↓

double f(double d1) {
        double r61317431 = d1;
        double r61317432 = 10.0;
        double r61317433 = pow(r61317431, r61317432);
        return r61317433;
}

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

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

4. Applied pow1 0.1

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

5. Applied pow-sqr 0.1

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

6. Applied pow1 0.1

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

7. Applied pow1 0.1

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

8. Applied pow-prod-up 0.1

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

9. Applied pow1 0.1

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

10. Applied pow-prod-up 0.1

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

11. Applied pow-plus 0.1

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

12. Applied pow-plus 0.1

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

13. Applied pow-prod-up 0.1

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

14. Applied pow-plus 0.1

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

15. Applied pow1 0.1

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

16. Applied pow-prod-up 0.0

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

17. Simplified 0.0

    \[\leadsto {d1}^{\color{blue}{9}} \cdot d1\]
18. Using strategy rm

19. Applied pow1 0.0

    \[\leadsto {d1}^{9} \cdot \color{blue}{{d1}^{1}}\]

20. Applied pow-prod-up 0

    \[\leadsto \color{blue}{{d1}^{\left(9 + 1\right)}}\]

21. Simplified 0

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

22. Final simplification 0

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

Reproduce

herbie shell --seed 2019128 
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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