Expression, p6

Average Error: 3.6 → 2.7

Time: 11.9s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r1485448 = a;
        double r1485449 = b;
        double r1485450 = c;
        double r1485451 = d;
        double r1485452 = r1485450 + r1485451;
        double r1485453 = r1485449 + r1485452;
        double r1485454 = r1485448 + r1485453;
        double r1485455 = 2.0;
        double r1485456 = r1485454 * r1485455;
        return r1485456;
}

↓

double f(double a, double b, double c, double d) {
        double r1485457 = 2.0;
        double r1485458 = d;
        double r1485459 = b;
        double r1485460 = c;
        double r1485461 = r1485459 + r1485460;
        double r1485462 = a;
        double r1485463 = r1485461 + r1485462;
        double r1485464 = r1485458 + r1485463;
        double r1485465 = r1485461 + r1485458;
        double r1485466 = r1485465 + r1485462;
        double r1485467 = r1485466 * r1485466;
        double r1485468 = r1485467 * r1485466;
        double r1485469 = cbrt(r1485468);
        double r1485470 = r1485469 * r1485466;
        double r1485471 = r1485464 * r1485470;
        double r1485472 = cbrt(r1485471);
        double r1485473 = r1485457 * r1485472;
        return r1485473;
}

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

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

9. Applied add-cbrt-cube 2.7

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

10. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019153 
(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))