Expression, p6

Average Error: 3.6 → 2.9

Time: 3.7s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r80358 = a;
        double r80359 = b;
        double r80360 = c;
        double r80361 = d;
        double r80362 = r80360 + r80361;
        double r80363 = r80359 + r80362;
        double r80364 = r80358 + r80363;
        double r80365 = 2.0;
        double r80366 = r80364 * r80365;
        return r80366;
}

↓

double f(double a, double b, double c, double d) {
        double r80367 = a;
        double r80368 = b;
        double r80369 = c;
        double r80370 = r80368 + r80369;
        double r80371 = d;
        double r80372 = r80370 + r80371;
        double r80373 = r80367 + r80372;
        double r80374 = 3.0;
        double r80375 = pow(r80373, r80374);
        double r80376 = cbrt(r80375);
        double r80377 = pow(r80376, r80374);
        double r80378 = cbrt(r80377);
        double r80379 = pow(r80378, r80374);
        double r80380 = cbrt(r80379);
        double r80381 = 2.0;
        double r80382 = r80380 * r80381;
        return r80382;
}

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	2.9

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

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. Simplified 2.9

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

8. Applied add-cbrt-cube 2.9

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

9. Simplified 2.9

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

11. Applied add-cbrt-cube 2.9

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

12. Simplified 2.9

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

13. Final simplification 2.9

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :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))