Expression, p6

Average Error: 3.7 → 2.9

Time: 7.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r125073 = a;
        double r125074 = b;
        double r125075 = c;
        double r125076 = d;
        double r125077 = r125075 + r125076;
        double r125078 = r125074 + r125077;
        double r125079 = r125073 + r125078;
        double r125080 = 2.0;
        double r125081 = r125079 * r125080;
        return r125081;
}

↓

double f(double a, double b, double c, double d) {
        double r125082 = a;
        double r125083 = d;
        double r125084 = b;
        double r125085 = c;
        double r125086 = r125084 + r125085;
        double r125087 = r125083 + r125086;
        double r125088 = r125082 + r125087;
        double r125089 = 3.0;
        double r125090 = pow(r125088, r125089);
        double r125091 = cbrt(r125090);
        double r125092 = pow(r125091, r125089);
        double r125093 = cbrt(r125092);
        double r125094 = pow(r125093, r125089);
        double r125095 = cbrt(r125094);
        double r125096 = 2.0;
        double r125097 = r125095 * r125096;
        return r125097;
}

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	2.9

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

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+ 2.7

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

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

13. Final simplification 2.9

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

Reproduce

herbie shell --seed 2019350 +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))