FastMath test5

Average Error: 0.1 → 0

Time: 41.8s

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 r163031237 = d1;
        double r163031238 = r163031237 * r163031237;
        double r163031239 = r163031237 * r163031238;
        double r163031240 = r163031239 * r163031237;
        double r163031241 = r163031240 * r163031237;
        double r163031242 = r163031241 * r163031238;
        double r163031243 = r163031242 * r163031237;
        double r163031244 = r163031237 * r163031243;
        double r163031245 = r163031244 * r163031237;
        return r163031245;
}

↓

double f(double d1) {
        double r163031246 = d1;
        double r163031247 = 10.0;
        double r163031248 = pow(r163031246, r163031247);
        return r163031248;
}

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-sqr 0.1

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

7. Applied pow1 0.1

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

8. Applied pow1 0.1

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

9. Applied pow-plus 0.1

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

10. Applied pow1 0.1

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

11. Applied pow-prod-up 0.1

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

12. Applied pow-prod-up 0.1

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

13. Applied pow-plus 0.1

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

14. Applied pow-prod-up 0.1

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

15. Applied pow-prod-up 0.1

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

16. Simplified 0.1

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

17. Taylor expanded around -inf 0

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

18. Final simplification 0

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

Reproduce

herbie shell --seed 2019120 
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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