Average Error: 0.0 → 0.0
Time: 12.8s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[b \cdot b + \left(a + \left(b + b\right)\right) \cdot a\]
\left(a + b\right) \cdot \left(a + b\right)
↓
b \cdot b + \left(a + \left(b + b\right)\right) \cdot a
double f(double a, double b) {
        double r8427741 = a;
        double r8427742 = b;
        double r8427743 = r8427741 + r8427742;
        double r8427744 = r8427743 * r8427743;
        return r8427744;
}
↓
double f(double a, double b) {
        double r8427745 = b;
        double r8427746 = r8427745 * r8427745;
        double r8427747 = a;
        double r8427748 = r8427745 + r8427745;
        double r8427749 = r8427747 + r8427748;
        double r8427750 = r8427749 * r8427747;
        double r8427751 = r8427746 + r8427750;
        return r8427751;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right)\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  5. Applied *-commutative0.0

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

  6. Applied *-commutative0.0

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

  7. Applied distribute-lft-out0.0

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

  8. Applied distribute-lft-out0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5 a 10) (<= 0 b 0.001))

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

  (* (+ a b) (+ a b)))