Expression, p6

Average Error: 3.6 → 2.7

Time: 13.6s

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

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

↓

\[2 \cdot \sqrt[3]{\log \left(e^{\left(a + c\right) + \left(b + d\right)}\right) \cdot \left(\left(\left(\left(b + c\right) + d\right) + a\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right)}\]

C

double f(double a, double b, double c, double d) {
        double r1995972 = a;
        double r1995973 = b;
        double r1995974 = c;
        double r1995975 = d;
        double r1995976 = r1995974 + r1995975;
        double r1995977 = r1995973 + r1995976;
        double r1995978 = r1995972 + r1995977;
        double r1995979 = 2.0;
        double r1995980 = r1995978 * r1995979;
        return r1995980;
}

↓

double f(double a, double b, double c, double d) {
        double r1995981 = 2.0;
        double r1995982 = a;
        double r1995983 = c;
        double r1995984 = r1995982 + r1995983;
        double r1995985 = b;
        double r1995986 = d;
        double r1995987 = r1995985 + r1995986;
        double r1995988 = r1995984 + r1995987;
        double r1995989 = exp(r1995988);
        double r1995990 = log(r1995989);
        double r1995991 = r1995985 + r1995983;
        double r1995992 = r1995991 + r1995986;
        double r1995993 = r1995992 + r1995982;
        double r1995994 = r1995993 * r1995993;
        double r1995995 = r1995990 * r1995994;
        double r1995996 = cbrt(r1995995);
        double r1995997 = r1995981 * r1995996;
        return r1995997;
}

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	2.7

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

Derivation

1. Initial program 3.6

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

3. Applied associate-+r+ 2.7

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

7. Applied add-log-exp 2.9

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

8. Applied add-log-exp 2.9

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

9. Applied add-log-exp 2.9

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

10. Applied sum-log 2.8

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

11. Applied sum-log 2.7

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

12. Applied add-log-exp 2.7

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

13. Applied sum-log 2.6

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

14. Simplified 2.7

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

16. Applied associate-+l+ 2.7

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

17. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019128 
(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))