Average Error: 3.6 → 2.9
Time: 3.8s
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}\right)}^{3}}\right)}^{3}} \cdot 2\]
\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
double f(double a, double b, double c, double d) {
        double r104627 = a;
        double r104628 = b;
        double r104629 = c;
        double r104630 = d;
        double r104631 = r104629 + r104630;
        double r104632 = r104628 + r104631;
        double r104633 = r104627 + r104632;
        double r104634 = 2.0;
        double r104635 = r104633 * r104634;
        return r104635;
}
↓
double f(double a, double b, double c, double d) {
        double r104636 = a;
        double r104637 = b;
        double r104638 = c;
        double r104639 = r104637 + r104638;
        double r104640 = d;
        double r104641 = r104639 + r104640;
        double r104642 = r104636 + r104641;
        double r104643 = 3.0;
        double r104644 = pow(r104642, r104643);
        double r104645 = cbrt(r104644);
        double r104646 = pow(r104645, r104643);
        double r104647 = cbrt(r104646);
        double r104648 = pow(r104647, r104643);
        double r104649 = cbrt(r104648);
        double r104650 = 2.0;
        double r104651 = r104649 * r104650;
        return r104651;
}

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
Herbie2.9
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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-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 add-cbrt-cube2.9

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

  9. Simplified2.9

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

  11. Applied add-cbrt-cube2.9

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

  12. Simplified2.9

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

  13. Final simplification2.9

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

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))