Expression, p6

Average Error: 3.6 → 0

Time: 7.7s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r4865949 = a;
        double r4865950 = b;
        double r4865951 = c;
        double r4865952 = d;
        double r4865953 = r4865951 + r4865952;
        double r4865954 = r4865950 + r4865953;
        double r4865955 = r4865949 + r4865954;
        double r4865956 = 2.0;
        double r4865957 = r4865955 * r4865956;
        return r4865957;
}

↓

double f(double a, double b, double c, double d) {
        double r4865958 = 2.0;
        double r4865959 = b;
        double r4865960 = c;
        double r4865961 = r4865959 + r4865960;
        double r4865962 = a;
        double r4865963 = d;
        double r4865964 = r4865962 + r4865963;
        double r4865965 = r4865961 + r4865964;
        double r4865966 = r4865958 * r4865965;
        return r4865966;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.6
Target	3.8
Herbie	0

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

Derivation

1. Initial program 3.6

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

3. Applied add-log-exp 3.6

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

4. Applied add-log-exp 3.6

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

5. Applied sum-log 3.6

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

6. Applied add-log-exp 3.6

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

7. Applied sum-log 2.8

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

8. Applied add-log-exp 2.8

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

9. Applied sum-log 1.5

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

10. Simplified 0.9

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

12. Applied exp-sum 1.2

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

13. Applied log-prod 0.0

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

14. Simplified 0.0

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

15. Simplified 0.0

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

17. Applied associate-+l+ 0

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

18. Final simplification 0

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

Reproduce

herbie shell --seed 2019164 
(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))