Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[56789 \le a \le 98765 \land 0.0 \le b \le 1 \land 0.0 \le c \le 0.0016773000000000001 \land 0.0 \le d \le 0.0016773000000000001\]
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[a \cdot \left(\left(c + d\right) + b\right)\]
a \cdot \left(\left(b + c\right) + d\right)
↓
a \cdot \left(\left(c + d\right) + b\right)
double f(double a, double b, double c, double d) {
        double r68880 = a;
        double r68881 = b;
        double r68882 = c;
        double r68883 = r68881 + r68882;
        double r68884 = d;
        double r68885 = r68883 + r68884;
        double r68886 = r68880 * r68885;
        return r68886;
}
↓
double f(double a, double b, double c, double d) {
        double r68887 = a;
        double r68888 = c;
        double r68889 = d;
        double r68890 = r68888 + r68889;
        double r68891 = b;
        double r68892 = r68890 + r68891;
        double r68893 = r68887 * r68892;
        return r68893;
}

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

Original0.0
Target0.0
Herbie0.0
\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

  1. Initial program 0.0

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

  3. Applied distribute-lft-in0.0

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

  5. Applied distribute-rgt-in0.0

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

  6. Applied associate-+l+0.0

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

  7. Simplified0.0

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

  8. Taylor expanded around inf 0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (<= 56789 a 98765) (<= 0.0 b 1) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

  (* a (+ (+ b c) d)))