Average Error: 0.4 → 0.3
Time: 3.4s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[a + \left(\left(e + \left(b + c\right)\right) + d\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
a + \left(\left(e + \left(b + c\right)\right) + d\right)
double f(double a, double b, double c, double d, double e) {
        double r104675 = e;
        double r104676 = d;
        double r104677 = r104675 + r104676;
        double r104678 = c;
        double r104679 = r104677 + r104678;
        double r104680 = b;
        double r104681 = r104679 + r104680;
        double r104682 = a;
        double r104683 = r104681 + r104682;
        return r104683;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r104684 = a;
        double r104685 = e;
        double r104686 = b;
        double r104687 = c;
        double r104688 = r104686 + r104687;
        double r104689 = r104685 + r104688;
        double r104690 = d;
        double r104691 = r104689 + r104690;
        double r104692 = r104684 + r104691;
        return r104692;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.2
Herbie0.3
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

    \[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied distribute-lft-out0.4

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

  6. Simplified0.4

    \[\leadsto \left(1 \cdot \color{blue}{\left(d + \left(e + c\right)\right)} + b\right) + a\]
  7. Using strategy rm

  8. Applied distribute-lft-in0.4

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

  9. Applied associate-+l+0.4

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

  10. Simplified0.4

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

  11. Taylor expanded around 0 0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020001 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))