FastMath test3

Average Error: 0.1 → 0.1

Time: 9.3s

Precision: 64

Math

\[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]

↓

\[\left(d3 + \left(3 + d2\right)\right) \cdot d1\]

C

double f(double d1, double d2, double d3) {
        double r219603 = d1;
        double r219604 = 3.0;
        double r219605 = r219603 * r219604;
        double r219606 = d2;
        double r219607 = r219603 * r219606;
        double r219608 = r219605 + r219607;
        double r219609 = d3;
        double r219610 = r219603 * r219609;
        double r219611 = r219608 + r219610;
        return r219611;
}

↓

double f(double d1, double d2, double d3) {
        double r219612 = d3;
        double r219613 = 3.0;
        double r219614 = d2;
        double r219615 = r219613 + r219614;
        double r219616 = r219612 + r219615;
        double r219617 = d1;
        double r219618 = r219616 * r219617;
        return r219618;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Target

Original	0.1
Target	0.1
Herbie	0.1

\[d1 \cdot \left(\left(3 + d2\right) + d3\right)\]

Derivation

1. Initial program 0.1

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]

2. Simplified 0.1

   \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(3 + d3\right)\right)}\]
3. Using strategy rm

4. Applied add-cube-cbrt 1.0

   \[\leadsto d1 \cdot \color{blue}{\left(\left(\sqrt[3]{d2 + \left(3 + d3\right)} \cdot \sqrt[3]{d2 + \left(3 + d3\right)}\right) \cdot \sqrt[3]{d2 + \left(3 + d3\right)}\right)}\]

5. Applied associate-*r* 1.1

   \[\leadsto \color{blue}{\left(d1 \cdot \left(\sqrt[3]{d2 + \left(3 + d3\right)} \cdot \sqrt[3]{d2 + \left(3 + d3\right)}\right)\right) \cdot \sqrt[3]{d2 + \left(3 + d3\right)}}\]

6. Simplified 1.2

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right)} \cdot \sqrt[3]{d2 + \left(3 + d3\right)}\]
7. Using strategy rm

8. Applied pow1 1.2

   \[\leadsto \left(\left(d1 \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \color{blue}{{\left(\sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}}\]

9. Applied pow1 1.2

   \[\leadsto \left(\left(d1 \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \color{blue}{{\left(\sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}\]

10. Applied pow1 1.2

    \[\leadsto \left(\left(d1 \cdot \color{blue}{{\left(\sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}\right) \cdot {\left(\sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}\]

11. Applied pow1 1.2

    \[\leadsto \left(\left(\color{blue}{{d1}^{1}} \cdot {\left(\sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}\right) \cdot {\left(\sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}\right) \cdot {\left(\sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}\]

12. Applied pow-prod-down 1.2

    \[\leadsto \left(\color{blue}{{\left(d1 \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}} \cdot {\left(\sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}\right) \cdot {\left(\sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}\]

13. Applied pow-prod-down 1.2

    \[\leadsto \color{blue}{{\left(\left(d1 \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right)}^{1}} \cdot {\left(\sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}\]

14. Applied pow-prod-down 1.2

    \[\leadsto \color{blue}{{\left(\left(\left(d1 \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \sqrt[3]{3 + \left(d3 + d2\right)}\right) \cdot \sqrt[3]{d2 + \left(3 + d3\right)}\right)}^{1}}\]

15. Simplified 0.1

    \[\leadsto {\color{blue}{\left(d1 \cdot \left(d3 + \left(d2 + 3\right)\right)\right)}}^{1}\]

16. Final simplification 0.1

    \[\leadsto \left(d3 + \left(3 + d2\right)\right) \cdot d1\]

Reproduce

herbie shell --seed 2019194 
(FPCore (d1 d2 d3)
  :name "FastMath test3"

  :herbie-target
  (* d1 (+ (+ 3.0 d2) d3))

  (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))