FastMath test5

Average Error: 0.1 → 0

Time: 43.5s

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 r3011147 = d1;
        double r3011148 = r3011147 * r3011147;
        double r3011149 = r3011147 * r3011148;
        double r3011150 = r3011149 * r3011147;
        double r3011151 = r3011150 * r3011147;
        double r3011152 = r3011151 * r3011148;
        double r3011153 = r3011152 * r3011147;
        double r3011154 = r3011147 * r3011153;
        double r3011155 = r3011154 * r3011147;
        return r3011155;
}

↓

double f(double d1) {
        double r3011156 = d1;
        double r3011157 = 10.0;
        double r3011158 = pow(r3011156, r3011157);
        return r3011158;
}

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

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

4. Applied pow1 0.1

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

5. Applied pow1 0.1

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

6. Applied pow1 0.1

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

7. Applied pow-prod-up 0.1

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

8. Applied pow1 0.1

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

9. Applied pow2 0.1

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

10. Applied pow1 0.1

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

11. Applied pow-prod-up 0.1

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

12. Applied pow-plus 0.1

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

13. Applied pow-prod-up 0.1

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

14. Applied pow-prod-up 0.1

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

15. Applied pow-prod-up 0.1

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

16. Applied pow1 0.1

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

17. Applied pow-prod-up 0.0

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

18. Applied pow-prod-up 0

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

19. Simplified 0

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

20. Final simplification 0

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (d1)
  :name "FastMath test5"

  :herbie-target
  (pow d1 10)

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