Expression, p6

Average Error: 3.7 → 2.7

Time: 12.1s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r1934120 = a;
        double r1934121 = b;
        double r1934122 = c;
        double r1934123 = d;
        double r1934124 = r1934122 + r1934123;
        double r1934125 = r1934121 + r1934124;
        double r1934126 = r1934120 + r1934125;
        double r1934127 = 2.0;
        double r1934128 = r1934126 * r1934127;
        return r1934128;
}

↓

double f(double a, double b, double c, double d) {
        double r1934129 = 2.0;
        double r1934130 = b;
        double r1934131 = c;
        double r1934132 = r1934130 + r1934131;
        double r1934133 = a;
        double r1934134 = r1934132 + r1934133;
        double r1934135 = d;
        double r1934136 = r1934134 + r1934135;
        double r1934137 = r1934136 * r1934136;
        double r1934138 = r1934136 * r1934137;
        double r1934139 = cbrt(r1934138);
        double r1934140 = r1934132 + r1934135;
        double r1934141 = r1934140 + r1934133;
        double r1934142 = r1934141 * r1934141;
        double r1934143 = r1934139 * r1934142;
        double r1934144 = cbrt(r1934143);
        double r1934145 = r1934129 * r1934144;
        return r1934145;
}

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	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 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+ 2.8

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

9. Applied add-cbrt-cube 2.7

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

10. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019156 
(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))