FastMath test5

Average Error: 0.1 → 0

Time: 13.2s

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 r8559729 = d1;
        double r8559730 = r8559729 * r8559729;
        double r8559731 = r8559729 * r8559730;
        double r8559732 = r8559731 * r8559729;
        double r8559733 = r8559732 * r8559729;
        double r8559734 = r8559733 * r8559730;
        double r8559735 = r8559734 * r8559729;
        double r8559736 = r8559729 * r8559735;
        double r8559737 = r8559736 * r8559729;
        return r8559737;
}

↓

double f(double d1) {
        double r8559738 = d1;
        double r8559739 = 10.0;
        double r8559740 = pow(r8559738, r8559739);
        return r8559740;
}

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 pow2 0.1

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

7. Applied pow1 0.1

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

8. Applied pow-prod-up 0.1

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

9. Applied pow-plus 0.1

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

10. Applied pow-plus 0.1

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

11. Applied pow-prod-up 0.1

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

12. Simplified 0.1

    \[\leadsto \left(d1 \cdot \left({d1}^{\color{blue}{7}} \cdot d1\right)\right) \cdot d1\]
13. Using strategy rm

14. Applied pow1 0.1

    \[\leadsto \left(d1 \cdot \left({d1}^{7} \cdot d1\right)\right) \cdot \color{blue}{{d1}^{1}}\]

15. Applied pow-plus 0.1

    \[\leadsto \left(d1 \cdot \color{blue}{{d1}^{\left(7 + 1\right)}}\right) \cdot {d1}^{1}\]

16. Applied pow1 0.1

    \[\leadsto \left(\color{blue}{{d1}^{1}} \cdot {d1}^{\left(7 + 1\right)}\right) \cdot {d1}^{1}\]

17. Applied pow-prod-up 0.0

    \[\leadsto \color{blue}{{d1}^{\left(1 + \left(7 + 1\right)\right)}} \cdot {d1}^{1}\]

18. Applied pow-prod-up 0

    \[\leadsto \color{blue}{{d1}^{\left(\left(1 + \left(7 + 1\right)\right) + 1\right)}}\]

19. Simplified 0

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

20. Final simplification 0

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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