FastMath test5

Average Error: 0.1 → 0

Time: 42.5s

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 r66205921 = d1;
        double r66205922 = r66205921 * r66205921;
        double r66205923 = r66205921 * r66205922;
        double r66205924 = r66205923 * r66205921;
        double r66205925 = r66205924 * r66205921;
        double r66205926 = r66205925 * r66205922;
        double r66205927 = r66205926 * r66205921;
        double r66205928 = r66205921 * r66205927;
        double r66205929 = r66205928 * r66205921;
        return r66205929;
}

↓

double f(double d1) {
        double r66205930 = d1;
        double r66205931 = 10.0;
        double r66205932 = pow(r66205930, r66205931);
        return r66205932;
}

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 pow2 0.1

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

5. Applied pow1 0.1

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

6. Applied pow1 0.1

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

7. Applied pow-sqr 0.1

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

8. Applied pow1 0.1

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

9. Applied pow-prod-up 0.1

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

10. Applied pow-plus 0.1

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

11. Applied pow-plus 0.1

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

12. Applied pow-prod-up 0.1

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

13. Applied pow-prod-up 0.1

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

14. Simplified 0.1

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

15. Taylor expanded around inf 0

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

16. Final simplification 0

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

Reproduce

herbie shell --seed 2019119 
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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