Average Error: 3.6 → 2.7
Time: 2.6s
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
2 \cdot \left(a + \left(\left(b + c\right) + d\right)\right)
double f(double a, double b, double c, double d) {
        double r82184 = a;
        double r82185 = b;
        double r82186 = c;
        double r82187 = d;
        double r82188 = r82186 + r82187;
        double r82189 = r82185 + r82188;
        double r82190 = r82184 + r82189;
        double r82191 = 2.0;
        double r82192 = r82190 * r82191;
        return r82192;
}
↓
double f(double a, double b, double c, double d) {
        double r82193 = 2.0;
        double r82194 = a;
        double r82195 = b;
        double r82196 = c;
        double r82197 = r82195 + r82196;
        double r82198 = d;
        double r82199 = r82197 + r82198;
        double r82200 = r82194 + r82199;
        double r82201 = r82193 * r82200;
        return r82201;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.6
Target3.8
Herbie2.7
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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+2.7

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

  5. Applied add-cbrt-cube2.9

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

  6. Simplified2.9

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

  8. Applied *-un-lft-identity2.9

    \[\leadsto \sqrt[3]{{\color{blue}{\left(1 \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right)}}^{3}} \cdot 2\]

  9. Applied unpow-prod-down2.9

    \[\leadsto \sqrt[3]{\color{blue}{{1}^{3} \cdot {\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}} \cdot 2\]

  10. Applied cbrt-prod2.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{{1}^{3}} \cdot \sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}\right)} \cdot 2\]

  11. Simplified2.9

    \[\leadsto \left(\color{blue}{1} \cdot \sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}\right) \cdot 2\]

  12. Simplified2.7

    \[\leadsto \left(1 \cdot \color{blue}{\left(a + \left(\left(b + c\right) + d\right)\right)}\right) \cdot 2\]

  13. Final simplification2.7

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

Reproduce

herbie shell --seed 2020057 +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))