Expression, p6

Average Error: 3.6 → 2.6

Time: 14.4s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r3690817 = a;
        double r3690818 = b;
        double r3690819 = c;
        double r3690820 = d;
        double r3690821 = r3690819 + r3690820;
        double r3690822 = r3690818 + r3690821;
        double r3690823 = r3690817 + r3690822;
        double r3690824 = 2.0;
        double r3690825 = r3690823 * r3690824;
        return r3690825;
}

↓

double f(double a, double b, double c, double d) {
        double r3690826 = d;
        double r3690827 = a;
        double r3690828 = c;
        double r3690829 = r3690827 + r3690828;
        double r3690830 = b;
        double r3690831 = r3690829 + r3690830;
        double r3690832 = r3690826 + r3690831;
        double r3690833 = exp(r3690832);
        double r3690834 = log(r3690833);
        double r3690835 = r3690830 + r3690828;
        double r3690836 = r3690835 + r3690827;
        double r3690837 = r3690826 + r3690836;
        double r3690838 = r3690834 * r3690837;
        double r3690839 = r3690835 + r3690826;
        double r3690840 = r3690839 + r3690827;
        double r3690841 = r3690838 * r3690840;
        double r3690842 = cbrt(r3690841);
        double r3690843 = 2.0;
        double r3690844 = r3690842 * r3690843;
        return r3690844;
}

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	2.6

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

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

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

8. Applied add-log-exp 2.9

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

9. Applied add-log-exp 2.9

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

10. Applied sum-log 2.8

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

11. Applied sum-log 2.7

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

12. Applied add-log-exp 2.7

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

13. Applied sum-log 2.6

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

14. Simplified 2.7

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

16. Applied associate-+r+ 2.6

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

17. Final simplification 2.6

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

Reproduce

herbie shell --seed 2019129 
(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))