Expression, p6

Average Error: 3.6 → 2.7

Time: 49.5s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r12779454 = a;
        double r12779455 = b;
        double r12779456 = c;
        double r12779457 = d;
        double r12779458 = r12779456 + r12779457;
        double r12779459 = r12779455 + r12779458;
        double r12779460 = r12779454 + r12779459;
        double r12779461 = 2.0;
        double r12779462 = r12779460 * r12779461;
        return r12779462;
}

↓

double f(double a, double b, double c, double d) {
        double r12779463 = c;
        double r12779464 = b;
        double r12779465 = a;
        double r12779466 = r12779464 + r12779465;
        double r12779467 = d;
        double r12779468 = r12779466 + r12779467;
        double r12779469 = r12779463 + r12779468;
        double r12779470 = exp(r12779469);
        double r12779471 = log(r12779470);
        double r12779472 = r12779464 + r12779463;
        double r12779473 = r12779472 + r12779467;
        double r12779474 = r12779473 + r12779465;
        double r12779475 = r12779472 + r12779465;
        double r12779476 = r12779475 + r12779467;
        double r12779477 = r12779474 * r12779476;
        double r12779478 = r12779471 * r12779477;
        double r12779479 = cbrt(r12779478);
        double r12779480 = 2.0;
        double r12779481 = r12779479 * r12779480;
        return r12779481;
}

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+ 2.8

   \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
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 associate-+r+ 2.7

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

9. Applied add-log-exp 2.7

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

10. Applied add-log-exp 2.7

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

11. Applied add-log-exp 2.7

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

12. Applied sum-log 2.7

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

13. Applied sum-log 2.7

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

14. Applied add-log-exp 2.7

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

15. Applied sum-log 2.3

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

16. Simplified 2.7

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

17. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019121 
(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))