Average Error: 3.6 → 2.7
Time: 1.5m
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \sqrt[3]{\log \left(e^{\left(a + c\right) + \left(b + d\right)}\right) \cdot \left(\left(\left(\left(b + c\right) + d\right) + a\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right)}\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
2 \cdot \sqrt[3]{\log \left(e^{\left(a + c\right) + \left(b + d\right)}\right) \cdot \left(\left(\left(\left(b + c\right) + d\right) + a\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right)}
double f(double a, double b, double c, double d) {
        double r42663408 = a;
        double r42663409 = b;
        double r42663410 = c;
        double r42663411 = d;
        double r42663412 = r42663410 + r42663411;
        double r42663413 = r42663409 + r42663412;
        double r42663414 = r42663408 + r42663413;
        double r42663415 = 2.0;
        double r42663416 = r42663414 * r42663415;
        return r42663416;
}
↓
double f(double a, double b, double c, double d) {
        double r42663417 = 2.0;
        double r42663418 = a;
        double r42663419 = c;
        double r42663420 = r42663418 + r42663419;
        double r42663421 = b;
        double r42663422 = d;
        double r42663423 = r42663421 + r42663422;
        double r42663424 = r42663420 + r42663423;
        double r42663425 = exp(r42663424);
        double r42663426 = log(r42663425);
        double r42663427 = r42663421 + r42663419;
        double r42663428 = r42663427 + r42663422;
        double r42663429 = r42663428 + r42663418;
        double r42663430 = r42663429 * r42663429;
        double r42663431 = r42663426 * r42663430;
        double r42663432 = cbrt(r42663431);
        double r42663433 = r42663417 * r42663432;
        return r42663433;
}

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. Using strategy rm

  7. Applied associate-+r+2.7

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

  9. Applied add-log-exp2.7

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

  10. Applied add-log-exp2.7

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

  11. Applied sum-log2.8

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

  12. Simplified2.7

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

  13. Final simplification2.7

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

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c d)
  :name "Expression, p6"
  :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))