Expression, p6

Average Error: 3.7 → 0

Time: 8.9s

Precision: 64

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

↓

\[2 \cdot \left(\left(b + c\right) + \left(d + a\right)\right)\]

C

double f(double a, double b, double c, double d) {
        double r79222 = a;
        double r79223 = b;
        double r79224 = c;
        double r79225 = d;
        double r79226 = r79224 + r79225;
        double r79227 = r79223 + r79226;
        double r79228 = r79222 + r79227;
        double r79229 = 2.0;
        double r79230 = r79228 * r79229;
        return r79230;
}

↓

double f(double a, double b, double c, double d) {
        double r79231 = 2.0;
        double r79232 = b;
        double r79233 = c;
        double r79234 = r79232 + r79233;
        double r79235 = d;
        double r79236 = a;
        double r79237 = r79235 + r79236;
        double r79238 = r79234 + r79237;
        double r79239 = r79231 * r79238;
        return r79239;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.8
Herbie	0

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

1. Initial program 3.7

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
2. Using strategy rm

3. Applied associate-+r+ 2.8

   \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
4. Using strategy rm

5. Applied add-cbrt-cube 2.9

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot 2\]

6. Simplified 2.9

   \[\leadsto \sqrt[3]{\color{blue}{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}} \cdot 2\]
7. Using strategy rm

8. Applied add-cbrt-cube 2.9

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}\right)}}^{3}} \cdot 2\]

9. Simplified 2.9

   \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left(\left(d + \left(b + c\right)\right) + a\right)}^{3}}}\right)}^{3}} \cdot 2\]
10. Using strategy rm

11. Applied *-un-lft-identity 2.9

    \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\color{blue}{\left(1 \cdot \left(\left(d + \left(b + c\right)\right) + a\right)\right)}}^{3}}\right)}^{3}} \cdot 2\]

12. Applied unpow-prod-down 2.9

    \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{1}^{3} \cdot {\left(\left(d + \left(b + c\right)\right) + a\right)}^{3}}}\right)}^{3}} \cdot 2\]

13. Applied cbrt-prod 2.9

    \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt[3]{{1}^{3}} \cdot \sqrt[3]{{\left(\left(d + \left(b + c\right)\right) + a\right)}^{3}}\right)}}^{3}} \cdot 2\]

14. Applied unpow-prod-down 2.9

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{{1}^{3}}\right)}^{3} \cdot {\left(\sqrt[3]{{\left(\left(d + \left(b + c\right)\right) + a\right)}^{3}}\right)}^{3}}} \cdot 2\]

15. Applied cbrt-prod 2.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{{1}^{3}}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\left(d + \left(b + c\right)\right) + a\right)}^{3}}\right)}^{3}}\right)} \cdot 2\]

16. Simplified 2.9

    \[\leadsto \left(\color{blue}{1} \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\left(d + \left(b + c\right)\right) + a\right)}^{3}}\right)}^{3}}\right) \cdot 2\]

17. Simplified 0

    \[\leadsto \left(1 \cdot \color{blue}{\left(\left(b + c\right) + \left(d + a\right)\right)}\right) \cdot 2\]

18. Final simplification 0

    \[\leadsto 2 \cdot \left(\left(b + c\right) + \left(d + a\right)\right)\]

Reproduce

herbie shell --seed 2019209 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))