Expression, p6

Average Error: 3.7 → 0.6

Time: 16.7s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r5670526 = a;
        double r5670527 = b;
        double r5670528 = c;
        double r5670529 = d;
        double r5670530 = r5670528 + r5670529;
        double r5670531 = r5670527 + r5670530;
        double r5670532 = r5670526 + r5670531;
        double r5670533 = 2.0;
        double r5670534 = r5670532 * r5670533;
        return r5670534;
}

↓

double f(double a, double b, double c, double d) {
        double r5670535 = d;
        double r5670536 = a;
        double r5670537 = r5670535 + r5670536;
        double r5670538 = c;
        double r5670539 = b;
        double r5670540 = r5670538 + r5670539;
        double r5670541 = r5670537 + r5670540;
        double r5670542 = r5670541 * r5670541;
        double r5670543 = r5670542 * r5670541;
        double r5670544 = cbrt(r5670543);
        double r5670545 = 2.0;
        double r5670546 = r5670544 * r5670545;
        return r5670546;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.9
Herbie	0.6

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

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 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 2.8

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

10. Simplified 0.6

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

11. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14.0 a -13.0) (<= -3.0 b -2.0) (<= 3.0 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))