Expression, p6

Average Error: 3.7 → 2.7

Time: 31.6s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r14534562 = a;
        double r14534563 = b;
        double r14534564 = c;
        double r14534565 = d;
        double r14534566 = r14534564 + r14534565;
        double r14534567 = r14534563 + r14534566;
        double r14534568 = r14534562 + r14534567;
        double r14534569 = 2.0;
        double r14534570 = r14534568 * r14534569;
        return r14534570;
}

↓

double f(double a, double b, double c, double d) {
        double r14534571 = 2.0;
        double r14534572 = b;
        double r14534573 = d;
        double r14534574 = r14534572 + r14534573;
        double r14534575 = a;
        double r14534576 = c;
        double r14534577 = r14534575 + r14534576;
        double r14534578 = r14534574 + r14534577;
        double r14534579 = exp(r14534578);
        double r14534580 = log(r14534579);
        double r14534581 = r14534572 + r14534576;
        double r14534582 = r14534581 + r14534575;
        double r14534583 = r14534573 + r14534582;
        double r14534584 = r14534583 * r14534583;
        double r14534585 = r14534580 * r14534584;
        double r14534586 = cbrt(r14534585);
        double r14534587 = r14534571 * r14534586;
        return r14534587;
}

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	2.7

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

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 associate-+r+ 2.8

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

7. Applied add-cbrt-cube 2.9

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

9. Applied add-log-exp 2.9

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

10. Applied add-log-exp 2.9

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

11. Applied add-log-exp 2.9

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

12. Applied sum-log 2.8

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

13. Applied add-log-exp 2.8

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

14. Applied sum-log 2.7

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

15. Applied sum-log 2.6

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

16. Simplified 2.7

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

17. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019124 +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))