Expression, p6

Average Error: 3.6 → 2.9

Time: 3.5s

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 r62369 = a;
        double r62370 = b;
        double r62371 = c;
        double r62372 = d;
        double r62373 = r62371 + r62372;
        double r62374 = r62370 + r62373;
        double r62375 = r62369 + r62374;
        double r62376 = 2.0;
        double r62377 = r62375 * r62376;
        return r62377;
}

↓

double f(double a, double b, double c, double d) {
        double r62378 = a;
        double r62379 = b;
        double r62380 = c;
        double r62381 = r62379 + r62380;
        double r62382 = d;
        double r62383 = r62381 + r62382;
        double r62384 = r62378 + r62383;
        double r62385 = 3.0;
        double r62386 = pow(r62384, r62385);
        double r62387 = cbrt(r62386);
        double r62388 = pow(r62387, r62385);
        double r62389 = cbrt(r62388);
        double r62390 = pow(r62389, r62385);
        double r62391 = cbrt(r62390);
        double r62392 = 2.0;
        double r62393 = r62391 * r62392;
        return r62393;
}

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 2020025 +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))