Expression, p6

Average Error: 3.7 → 2.7

Time: 12.0s

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 r1690256 = a;
        double r1690257 = b;
        double r1690258 = c;
        double r1690259 = d;
        double r1690260 = r1690258 + r1690259;
        double r1690261 = r1690257 + r1690260;
        double r1690262 = r1690256 + r1690261;
        double r1690263 = 2.0;
        double r1690264 = r1690262 * r1690263;
        return r1690264;
}

↓

double f(double a, double b, double c, double d) {
        double r1690265 = 2.0;
        double r1690266 = a;
        double r1690267 = b;
        double r1690268 = c;
        double r1690269 = r1690267 + r1690268;
        double r1690270 = d;
        double r1690271 = r1690269 + r1690270;
        double r1690272 = r1690266 + r1690271;
        double r1690273 = r1690272 * r1690272;
        double r1690274 = r1690273 * r1690272;
        double r1690275 = cbrt(r1690274);
        double r1690276 = r1690269 + r1690266;
        double r1690277 = r1690270 + r1690276;
        double r1690278 = r1690275 * r1690277;
        double r1690279 = r1690278 * r1690272;
        double r1690280 = cbrt(r1690279);
        double r1690281 = r1690265 * r1690280;
        return r1690281;
}

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

   \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\left(a + \left(b + c\right)\right) + d\right)} \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]
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 2019154 
(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))