FastMath test5

Average Error: 0.1 → 0

Time: 42.2s

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 r4762650 = d1;
        double r4762651 = r4762650 * r4762650;
        double r4762652 = r4762650 * r4762651;
        double r4762653 = r4762652 * r4762650;
        double r4762654 = r4762653 * r4762650;
        double r4762655 = r4762654 * r4762651;
        double r4762656 = r4762655 * r4762650;
        double r4762657 = r4762650 * r4762656;
        double r4762658 = r4762657 * r4762650;
        return r4762658;
}

↓

double f(double d1) {
        double r4762659 = d1;
        double r4762660 = 10.0;
        double r4762661 = pow(r4762659, r4762660);
        return r4762661;
}

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-prod-up 0.1

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

6. Applied pow1 0.1

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

7. Applied pow1 0.1

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

8. Applied pow1 0.1

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

9. Applied pow-sqr 0.1

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

10. Applied pow1 0.1

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

11. Applied pow-prod-up 0.1

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

12. Applied pow-plus 0.1

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

13. Applied pow-prod-up 0.1

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

14. Applied pow-prod-up 0.1

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

15. Applied pow-plus 0.1

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

16. Applied pow1 0.1

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

17. Applied pow-prod-up 0.0

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

18. Applied pow-plus 0

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

19. Simplified 0

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

20. Final simplification 0

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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