Expression, p6

Average Error: 3.7 → 0

Time: 8.4s

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 r3651054 = a;
        double r3651055 = b;
        double r3651056 = c;
        double r3651057 = d;
        double r3651058 = r3651056 + r3651057;
        double r3651059 = r3651055 + r3651058;
        double r3651060 = r3651054 + r3651059;
        double r3651061 = 2.0;
        double r3651062 = r3651060 * r3651061;
        return r3651062;
}

↓

double f(double a, double b, double c, double d) {
        double r3651063 = 2.0;
        double r3651064 = b;
        double r3651065 = c;
        double r3651066 = r3651064 + r3651065;
        double r3651067 = d;
        double r3651068 = a;
        double r3651069 = r3651067 + r3651068;
        double r3651070 = r3651066 + r3651069;
        double r3651071 = r3651063 * r3651070;
        return r3651071;
}

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-log-exp 2.8

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

6. Applied add-log-exp 2.8

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

7. Applied add-log-exp 2.8

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

8. Applied sum-log 2.8

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

9. Applied sum-log 2.8

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

10. Applied add-log-exp 2.8

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

11. Applied sum-log 1.5

    \[\leadsto \color{blue}{\log \left(e^{a} \cdot \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)\right)} \cdot 2\]

12. Simplified 0.9

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

14. Applied exp-sum 1.2

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

15. Applied log-prod 0.8

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

16. Simplified 0.5

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

17. Simplified 0

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

18. Final simplification 0

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p6"
  :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))