Expression, p6

Average Error: 3.7 → 2.7

Time: 16.6s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r1751122 = a;
        double r1751123 = b;
        double r1751124 = c;
        double r1751125 = d;
        double r1751126 = r1751124 + r1751125;
        double r1751127 = r1751123 + r1751126;
        double r1751128 = r1751122 + r1751127;
        double r1751129 = 2.0;
        double r1751130 = r1751128 * r1751129;
        return r1751130;
}

↓

double f(double a, double b, double c, double d) {
        double r1751131 = 2.0;
        double r1751132 = a;
        double r1751133 = b;
        double r1751134 = c;
        double r1751135 = r1751133 + r1751134;
        double r1751136 = d;
        double r1751137 = r1751135 + r1751136;
        double r1751138 = r1751132 + r1751137;
        double r1751139 = r1751138 * r1751138;
        double r1751140 = r1751139 * r1751138;
        double r1751141 = cbrt(r1751140);
        double r1751142 = r1751135 + r1751132;
        double r1751143 = r1751136 + r1751142;
        double r1751144 = r1751141 * r1751143;
        double r1751145 = r1751144 * r1751138;
        double r1751146 = cbrt(r1751145);
        double r1751147 = r1751131 * r1751146;
        return r1751147;
}

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 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

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

9. Applied add-cbrt-cube 2.7

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

10. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019151 
(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))