Expression, p6

Average Error: 3.7 → 2.5

Time: 32.0s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r27931698 = a;
        double r27931699 = b;
        double r27931700 = c;
        double r27931701 = d;
        double r27931702 = r27931700 + r27931701;
        double r27931703 = r27931699 + r27931702;
        double r27931704 = r27931698 + r27931703;
        double r27931705 = 2.0;
        double r27931706 = r27931704 * r27931705;
        return r27931706;
}

↓

double f(double a, double b, double c, double d) {
        double r27931707 = 2.0;
        double r27931708 = b;
        double r27931709 = exp(r27931708);
        double r27931710 = log(r27931709);
        double r27931711 = a;
        double r27931712 = d;
        double r27931713 = r27931711 + r27931712;
        double r27931714 = c;
        double r27931715 = r27931713 + r27931714;
        double r27931716 = r27931710 + r27931715;
        double r27931717 = r27931708 + r27931714;
        double r27931718 = r27931717 + r27931712;
        double r27931719 = r27931718 + r27931711;
        double r27931720 = r27931719 * r27931719;
        double r27931721 = r27931716 * r27931720;
        double r27931722 = cbrt(r27931721);
        double r27931723 = r27931707 * r27931722;
        return r27931723;
}

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	2.5

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

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. 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

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

9. Applied sum-log 2.7

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

10. Applied add-log-exp 2.7

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

11. Applied sum-log 2.6

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

12. Simplified 2.9

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

14. Applied exp-sum 2.9

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

15. Applied log-prod 2.9

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

16. Simplified 2.5

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

17. Final simplification 2.5

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

Reproduce

herbie shell --seed 2019107 
(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))